Superconducting qubit circuit emulation of a vector spin-1/2

We propose a superconducting qubit circuit that can fully emulate a quantum vector spin-1/2, with an effective dipole moment having three independent components whose operators obey the commutation relations of a vector angular momentum in the computational subspace. Each component couples to an independently-controllable external bias, emulating the Zeeman effect due to a fictitious, vector magnetic field, and all three of these vector components remain relatively constant over a broad range of emulated total fields around zero. This capability, combined with established techniques for qubit coupling, should enable for the first time the direct hardware emulation of nearly arbitrary quantum spin-1/2 systems, including the canonical Heisenberg model. Furthermore, it would constitute a crucial step both towards realizing the full potential of quantum annealing, as well as exploring important quantum information processing capabilities that have so far been inaccessible to available hardware, such as quantum error suppression, Hamiltonian and holonomic quantum computing, and adiabatic quantum chemistry.

Quantum spin-1/2 models serve as basic paradigms for a wide variety of physical systems in quantum statistical mechanics and many-body physics, and are among the most highly studied in the context of quantum phase transitions and topological order [1]. In addition, since the spin-1/2 in a magnetic field is one of the simplest realizations of a qubit, many quantum information processing paradigms draw heavily on concepts which originated from or are closely related to quantum magnetism. For example, quantum spin-1/2 language is used to describe nearly all of the constructions underlying quantum errorcorrection [2] and error-suppression [3,4] methods. It is also the most commonly-used framework for specifying engineered Hamiltonians in many other quantum protocols such as quantum annealing [5], adiabatic topological quantum computing [6], quantum simulation [7,8], Hamiltonian and holonomic quantum computing [4,9], and quantum chemistry [10].
The conventional method for simulating vector spin-1/2 Hamiltonians is based on the "gate-model" quantum simulation paradigm, and uses pulsed, high-speed sequences of discrete, non-commuting gate operations [11] to approximate time-evolution under a desired Hamiltonian [12]. In this paradigm, simulating a different Hamiltonian simply requires reprogramming the hardware with a different sequence of gates, a desirable property so long as the error introduced by the discretization can be kept sufficiently low. Unfortunately, this becomes increasingly difficult as the required spin interactions become stronger and/or more complex, since for a fixed gate duration [13], the discretization error grows both with the strength of the spin-spin interactions, and with the number of mutually-noncommuting terms they contain. In addition, gate-based implementation necessarily implies that the system occupies Hilbert space far above its ground state, and the resulting information exchange with the environment via both absorption and emission is at the root of the need for active quantum error correction [2].
In light of these considerations, static emulation of a desired Hamiltonian (which suffers from neither of the above problems) becomes a potentially appealing alternative for applications requiring strong, complex spinspin interactions. In fact, the resulting intrinsic "protection" from noise associated with remaining in the ground state of a quasi-static Hamiltonian is precisely the motivation for adiabatic quantum computation protocols [14]. To this end, a wide array of what might be called "weakly-engineered," but still fundamentally naturallyoccurring, quantum magnetic systems have been explored for their potential quantum information applications, including, for example: the doped ionic crystals used in the original demonstration of quantum annealing [15], ultracold atoms [16] or dipolar molecules [17], chemically-engineered molecular nanomagnets [18], and donor spins in Si [19]. These kind of systems, however, do not have the level of microscopic controllability required in most cases for the applications discussed above.
A game-changing step toward this level of controllability was taken with the advent of the superconducting machines from D-Wave systems [8,20,21], which are the first examples of large-scale, "fully-engineered" quantum spin model emulators, and have already been used to great effect in both quantum annealing [5] and quantum simulation [8]. However, even these systems have critical limitations in their ability to emulate spin interactions, which are inherited directly from the persistent-current flux qubits on which their spins are based. In particular, as we discuss in detail below, although these qubits are well-suited to emulation of the simpler, transverse-field Ising spin (which interacts with other spins only via its z-component), a true vector spin-1/2 is beyond their capabilities. A direct consequence of this is their inability to realize so-called "non-stoquastic" two-qubit interaction Hamiltonians [22], which have been the subject of intense research interest in recent years [23,24] due to their potentially critical role in maximizing the computational power of quantum annealing. In spite of this great potential, no experimental demonstration of such a capability has yet been made, simply because of the limitations of existing qubit hardware.

arXiv:1810.01352v3 [quant-ph] 23 Jan 2019
In this work, we propose a novel superconducting circuit called the "Josephson phase-slip qubit" (JPSQ), a device which aims for the first time to directly emulate a fully-controllable, quantum vector spin-1/2. When combined with existing techniques for coupling superconducting qubits, this circuit could enable the realization of nearly arbitrary, controllable many-body spin Hamiltonians, without the limitations associated with digital, gate-based quantum simulation methods. Such a capability would provide entirely new modes of access to all of the applications listed above, many of which have never been tried experimentally due to the lack of required capabilities in qubit hardware.
The remainder of the paper is divided as follows: In section I we begin by describing how persistent-current flux qubits can be used to emulate quantum spin models, and their fundamental limitations in this context. We then introduce in section II our proposed Josephson phase-slip qubit, analyzing a simplified version of the circuit in detail, and comparing the results with numerical simulations. This section concludes with a discussion of the coherence of the JPSQ in the context of existing superconducting qubit devices. Section III contains discussion and simulations of two multi-JPSQ circuit examples: (i) two JPSQs coupled by both zz and xx interactions, and (ii) a four-JPSQ circuit which implements a distance-2 Bacon-Shor logical qubit with quantum error suppression [3,4]. Finally, section IV describes a generalization of the JPSQ circuit capable of emulating fullycontrollable, anisotropic Heisenberg interactions, and we conclude in section V. Superconducting circuits are already among the most engineerable high-coherence quantum systems available, allowing a range of behavior and interactions to be constructed by design [29,30]. Their capability to emulate complex static spin Hamiltonians is exemplified by the flux-qubit-based machines of D-Wave Systems, Inc. [8,20,21]. These systems are designed to emulate the quantum transverse-field Ising model, with Hamiltonian: Each spin has an emulated local magnetic field in the x − z plane with effective Zeeman energies given by the parameters E z i and E x i in eq. 1, and pairwise couplings to other spins parametrized by the Ising interaction energies J ij . Notably absent from this Hamiltonian are any interactions between the transverse moments of the spins, whose physical realization is the subject of this work.
To better understand what follows, we first describe the physics underlying emulation of eq. 1 with 2-loop flux qubits [20,[25][26][27][28], and why these circuits cannot be used to emulate the vector spin interactions of interest here. Figure 1 illustrates how flux qubits are currently used to emulate quantum spins. The circuit shown in panel (a) is the basic 4-junction flux qubit [20,[25][26][27][28], having two loops biased by fluxes Φ z and Φ x , labelled according to the spin moment they are used to emulate [31]. When the z loop is biased with an external flux Φ z = Φ 0 /2+δΦ z (where Φ 0 ≡ h/2e is the superconducting fluxoid quantum, and δΦ z Φ 0 ), the two lowest-energy semiclassical states of the loop are nearly degenerate due to the Meissner effect, having approximately equal and opposite supercurrents. As shown in fig. 1(a), these two semiclassical states correspond to expulsion of the external flux from the loop, or pulling additional flux into it, such that it contains exactly zero or one fluxoid quantum, respectively. They can be identified with two local minima in a magnetic potential (panel (b) in the figure) experienced by a fictitious particle whose "position" corresponds to the gauge-invariant phase difference across the two larger Josephson junctions (which play the role of a loop inductance), and whose "momentum" corresponds to the total charge that has flowed through them. The difference in potential energy between these two minima is controlled by δΦ z (panel (d)), and approximately corresponds to the interaction energy between an applied field and the two equal and opposite persistent currents (panel (e)). These two states naturally play the role of | ± z , the eigenstates ofσ z for the emulated spin. Quantum coupling between these two states corresponds to the operatorσ x : the Zeeman Hamiltonian due to an emulated transverse field. As shown in (a)-(c), this can be viewed as tunneling of a "fluxon" between the two states in which it is inside or outside of the larger loop, with a strength controlled by the barrier height via Φ x (panel (c)).
In published experimental work to date (with the exception of ref. 24), this Φ x tuning has only been used to adjust the Zeeman energy due to an effective transverse field [20,[25][26][27][28]. However, we are concerned here with producing a static interaction between the transverse dipole moments of two such emulated spins.  24. The DC SQUID loops of the two qubits (whose external flux biases control the height of their fluxon tunnel barriers) are each coupled by a mutual inductance M to a common flux qubit coupler. The latter can be viewed semiclassically as a tunable effective inductance L eff C which can assume positive or negative values, depending on how it is biased [21,[33][34][35][36][37]; therefore, we can qualitatively understand the resulting two-qubit interaction using the Hamiltonian: FIG. 1. Spin-1/2 emulation using a two-loop flux qubit. Panel (a) illustrates the persistent current (fluxon) states of the two-loop flux qubit [20,[25][26][27][28]. On the left is the state of NΦ = 0 where the persistent current expels the externally-applied flux Φ z , while on the right is the state NΦ = 1 in which a persistent current in the opposite direction pulls in the additional flux needed to trap exactly one fluxon in the loop. The path connecting these two semiclassical persistent current states to each other traverses an energy barrier in which the fluxon is stored inside the junctions of the DC SQUID interrupting the loop, as illustrated in the center panel of (a). Panels (b) and (c) show how this can be viewed in terms of the motion of a fictitious "phase" particle in a double-well potential. In (b), the barrier is high, and the two persistent current states are well-isolated, resulting in the energy levels vs. flux shown in (d), which emulate an Ising moment in a z field. Panel (c) illustrates that when the barrier is lowered by threading a flux Φ x through the DC SQUID loop, quantum tunneling between the two fluxon states occurs, producing the energy levels in (f). The tunneling appears as an avoided crossing between the two fluxon states of (d), which emulates the effect of a transverse field along x. Panels (e) and (g) show the resulting effective persistent currents I z q (δΦ z ) and I x (Φ x ) for the cases where the emulated transverse field is small, and large, respectively. Finally, panel (h) illustrates the resulting fundamental asymmetry between emulated z and x fields for the 2-loop flux qubit system, for the parameters: EJa = h × 44.7 GHz, CJa = 1.80 fF, EJ = h × 134 GHz, CJ = 5.40 fF (note that the junction capacitances are not shown).
where the coupler ground state is described semiclassically in terms of its inductance L eff C and current I C , Φ x A , Φ x B are the static flux offsets applied to the two qubits' DC SQUID loops, and we describe each qubit q ∈ {A, B} in terms of its Φ x −dependent quantum eigenenergies (in the absence of coupling) at zero effective z field (Φ z = Φ 0 /2): To calculate the effective coupling energy J xx in eq. 2, we expand the qubit energies around the points Φ x A and Φ x B , and then minimize the total energy with respect to the coupler current I C (following ref. 39), to obtain: where we have defined the Taylor coefficients (with q ∈ {A, B}): and neglected the differential quantum inductance between the | ± x states. Equations 4 and 5 show that the semiclassical quantity which plays the role of the qubit magnetic moment along the fictitious x direction is I x q , the slope of the tunnel splitting energy ∆E x q with respect to Φ x . This quantity is plotted in fig. 2(b) in blue (right axis) as a function of Φ x for a single tunable flux qubit, using the full numerical simulation methods of ref. 38 (previously used in refs. 37 and 40). Panel (c) shows in black the corresponding transverse coupling energy J xx  [38]. Because of the exponential decrease of I x q with Φ x shown in (b), the coupling energy goes exponentially to zero with ∆E x , so that non-negligible transverse interaction can only be achieved in the presence of large offset x fields for both qubits.
obtained by plugging this into eq. 4. The crucial point here is that the xx coupling that can be achieved in this way is always much smaller than the local transversefield Zeeman energies: The physical reason for this is simple: the Φ x q -dependence of the tunneling energy ∆E x q is exponential, since increasing the flux corresponds to lowering the tunnel barrier [c.f., fig. 1(h)]. Since the effective x magnetic moment of each qubit [c.f., eq. 5] is the derivative with respect to Φ x of this energy, it can only be large when the energy is itself large. In spin language, this constraint on transverse coupling between flux qubits corresponds to the x magnetic moments of the emulated spins going exponentially to zero as their local x fields go to zero, as shown in fig. 1(g). This limitation renders present-day superconducting circuits unable to emulate (in the static fashion under discussion here) the majority of the spin models discussed in the introduction, with the notable exception of the transverse field Ising model of eq. 1 emulated by D-Wave systems' machines [8].

II. JOSEPHSON PHASE-SLIP QUBIT (JPSQ)
In section I, we described how emulation of a large x magnetic moment with a persistent-current qubit requires its tunneling energy to be sensitive to barrier height, which implies that there must be substantial probability inside the barrier. For this to remain true all the way to zero transverse field (tunneling energy), the wavefunction inside the barrier must remain appreciable even when the tunneling itself goes to zero, a selfcontradictory requirement that cannot be satisfied by conventional flux qubits. Figure 3 shows a persistentcurrent qubit circuit which can. Here, the tunable fluxon tunnel barrier that provides control of the transverse field (which in fig. 1(a) is realized with a DC SQUID) consists of two DC SQUIDs in series, separated by a central superconducting island whose polarization charge can be controlled with an external bias voltage. This object is similar to the so-called "quantum phase-slip transistor" (QPST) [41] (with the quantum phase-slip junctions here replaced by DC SQUIDs), hence the name "Josephson phase-slip qubit" (JPSQ). The key feature which motivates the use of a QPST to control fluxon tunneling in the present context is that it provides two fluxon tunneling paths into or out of the loop, as illustrated in (a). If a polarization charge Q b is present on the island, the two fluxon tunneling amplitudes acquire a relative phase shift due to the Aharonov-Casher effect [42]; when Q b = e this phase shift is π, and if the magnitudes of the two tunneling amplitudes are equal, total suppression of fluxon tunneling occurs and the transverse field Zeeman energy is zero [ fig. 3(c)]. Crucially, this remains true even when the individual tunneling amplitudes are large and fluxsensitive, allowing a strong, linear flux-sensitivity (magnetic dipole moment) to persist even around zero field. We note before proceeding that a number of previous works have highlighted and/or experimentally exploited this phenomenon as a means to observe the Aharonov-Casher effect in superconducting circuits [43]; here, we propose a way to use it to realize a superconductingcircuit-based vector spin-1/2 qubit.  where the persistent current expels the externally-applied flux Φ z , while on the right is the state NΦ = 1 in which a persistent current in the opposite direction pulls in the additional flux needed to trap one fluxon in the loop. Two paths connect these two semiclassical persistent current states to each other, each of which contains an energy barrier (in which the fluxon is stored inside the junctions of one of the two DC SQUIDs interrupting the loop) as illustrated in the center panel of (a). Panel (b) shows the simplified circuit considered in our analytic analysis. Panels (c)-(f) illustrate the fluxon tunneling amplitudes which create an effective transverse-field Zeeman splitting, under several different conditions: in (c), the island is polarized with an offset charge e, and the two DC SQUIDs are biased with flux of equal magnitude Φ∆, corresponding to zero effective field; in (d), the two DC SQUID fluxes are imbalanced such that the two fluxon tunneling amplitudes no longer cancel completely, corresponding to a nonzero effective ±x field; in (e) a nonzero z field is added; finally, in (f) the island polarization charge is displaced by ξ, resulting in a nonzero effective y field.
with an offset flux of the same magnitude Φ L = ±Φ R = Φ ∆ , so that their individual fluxon tunneling amplitudes A L (Φ ∆ ) and A R (±Φ ∆ ) have equal magnitude, and are flux-sensitive (the ± indicates that there are two possible choices for the relative sign). If we then magnetically couple an input flux δΦ x to both DC SQUIDs with equal strength, and signs such that the resulting total flux through the two DC SQUIDs is affected oppositely, the two tunneling amplitudes no longer cancel, creating an effective transverse field as illustrated in fig. 3(d). If we also change the flux δΦ z , the total effect is analogous to a field in between the x and z axes, as shown in Panel (e). Panel (f) shows the effect of charge displacements away from half a Cooper pair, which act as transverse fields in the y direction.

A. JPSQ Hamiltonian
We now validate the intuitive picture given above by analyzing the quantum mechanics of the JPSQ circuit. As we show below, this analysis can be used both to understand the physics of the circuit, and to make semi-quantitative predictions of its important properties. The classical Hamiltonian for the JPSQ circuit of fig. 3(b) can be written: where the three terms are: (i) the electrostatic energy; (ii) the interaction with a polarization charge Q b = (0, 0, Q b ) (supplied by a bias source); and (iii) the Josephson potential energy. We start by transforming to the following loop, island, and plasma mode phase coordinates: where here, and below, we use lowercase φ to indicate dimensionless flux (phase) quantities, according to: φ ≡ 2πΦ/Φ 0 . In this coordinate representation, the Josephson potential is, to first order in δΦ x : where we have defined the effective DC SQUID Josephson energyẼ Ja (φ ∆ ) ≡ 2E Ja cos (φ ∆ /2) and the ratio: between the Josephson inductances of the DC SQUIDs (the fluxon tunneling elements) and that of the larger loop junctions (functioning as the inductance appearing "across" them) [44]. In this circuit, just as in the case of 3-and 4-junction flux qubits, the plasma mode φ p (the symmetric oscillation across the two larger Josephson junctions), can in most cases of interest be treated as a "bystander," in the sense that the energy barrier between fluxon states along this direction, as well as its characteristic oscillation frequency, are both usually much larger than the energy scale of interest for the qubit, such that to a good approximation the φ p mode does not participate in relevant phenomena at that energy scale.
show the potential energy surface of eq. 9 obtained by setting φ p = 0, for δφ z = δφ x = 0. Whereas the essential physics of a conventional flux qubit can be approximately described in terms a phase particle moving in a one-dimensional double-well potential [c.f., figs. 1(b),(c)] [25], whose position corresponds to the gauge-invariant phase across the qubit's loop inductance, the JPSQ is fundamentally different in that at least two dynamical variables are needed to capture even its qualitative properties. At a very high level, this difference can be associated with the fact that Josephson symmetry (under translations of Φ 0 ) does not play an essential role in the important low-energy properties of the flux qubit, while it is essential to those of the JPSQ: for the flux qubit of fig. 1(a), if we neglect spurious fluxon tunneling through the larger, loop junctions, the only nontrivial tunneling path for a fluxon is the usual one connecting the two persistent current states. However, as shown in figs. 3 and 4, for the JPSQ there is a closed fluxon tunneling path which encircles the island, corresponding to a discrete Josephson symmetry. This symmetry is highlighted in fig. 4(b) by representing φ I (the phase of the island) as an angular coordinate [45]. Fig. 4(b) illustrates how the two fluxon tunneling paths between persistent current states shown in fig. 3(a) appear on the 2D potential surface, as red and blue arrows, and how they correspond to motion in the two angular directions along φ I . This coordinate can be viewed as the angular coordinate of a fluxon circling around the island (passing through the DC SQUIDs' junction barriers), one period of which corresponds to a voltage pulse on the island of area ±Φ 0 , with the sign determined by the direction of motion. When there is an offset charge on the island, the system becomes sensitive to this direction, via the second term of eq. 7; when the island is polarized with exactly half a Cooper pair, the resulting π relative phase shift produces the destructive interference shown in fig. 3(c). We will see below that this can also be viewed as a geometric phase shift.
The two local minima of eq. 9, illustrated in fig. 4(b), correspond semi-classically to the two persistent current states, and are given by (for φ z = π, and to first order in δφ x ): where we have defined: In the limit where quantum phase fluctuations about these classical minima and the tunneling between them are negligible, the average magnitude of the corresponding equal and opposite persistent supercurrents circulating in the loop can be written down directly: whereĨ Ca (φ ∆ ) ≡Ẽ Ja (φ ∆ )×2π/Φ 0 is the effective critical current of each DC SQUID. The corresponding result for the semi-classical persistent current of the flux qubit is: where the angle γ(φ ∆ ) is defined by . Equations 11 and 12 are plotted in fig. 5, and they exhibit a qualitative difference between the two circuits when viewed as persistent-current qubits. The flux qubit result is shown in red, and displays the well-known behavior that a double-well potential only exists when 0.5 < β < 1; that is, the Josephson inductance of the small junction must be less than that of the loop. Within this range, the persistent current varies strongly, having a peak at β = 1/ √ 2. By constrast, the JPSQ can have a well-defined classical persistent current for any value of β, and in fact the current asymptotically approachesĨ Ca as β gets larger; that is, as the loop inductance becomes negligible compared to that of the DC SQUIDs. Note that this is a regime where the flux qubit does not have a double-well potential at all.
This qualitative difference can be intuitively explained  fig. 3(b) obtained by setting φp = 0 (a very good approximation for the parameters of interest here). Its two coordinates are: φ l , the phase difference across the two larger loop junctions; and φI , the phase of the island. Since the Josephson barrier along φ l (for tunneling of fluxons through the two big junctions) is much larger than other energy scales, we neglect tunneling through it and focus on the region near φ l = 0, illustrated by the shaded box in (a). Panel (b) shows a contour plot of the potential in this reqion, where the Josephson translational symmetry along φI has been made explicit by treating it as an angular coordinate. Inside each unit cell of this potential (one angular period of φI ) there are two local minima, corresponding to the two persistent current states having equal and opposite values of φ l (indicated by dashed circles). Red and blue 3D arrows indicate the two separate paths that connect these two minima (associated with the two possible angular directions in φI ), which correspond to tunneling of a fluxon through one or the other of the two DC SQUIDs in fig. 3(b). Panels (c) and (d) compare the potential for the cases: δφ z = 0, δφ x = 0 and δφ z = 0, δφ x = 0, respectively. To facilitate this comparison, they are shown "unwound" along the φI coordinate, with two full periods visible, and the persistent current states labelled by horizontal dashed lines. From this perspective, the two tunneling paths correspond to inter-unit-cell and intra-unit-cell tunneling (the unit cell is indicated with gray shading). Panel (e) illustrates quantities defined in the text, for a single unit cell: thin, solid red and blue arrows illustrate the coordinates defined in eq. 20; thick, dashed, red and blue arrows the linear approximation to the extremal paths used in our analysis; and, thin black arrows the displacements along φI of the two potential minima when δx = 0. Panel (f) illustrates the approximate 1D potential, defined on a circle, obtained by evaluating the full potential on the paths shown in (d). Panel (g) shows the potentials for the two tunneling paths overlaid (solid lines -exact result from eq. 9, dashed lines -eq. 26), illustrating that when δx = 0, the barrier and the length of one path are increased while for the other they are decreased.
by examining the effective inductive division occurring in the two cases. For the JPSQ, the top of the potential barrier between persistent current states corresponds to a gauge-invariant phase difference of π/2 across each of the two DC SQUIDs, such that their effective Josephson inductances formally diverge. The SQUIDs therefore look approximately like two series current sources, supplying their critical currentĨ Ca (φ ∆ ) to a "load" consisting of the two larger junctions (as long as β is not close to 1). For the tunable flux qubit, however, the top of the potential barrier between persistent current states corresponds to a gauge-invariant phase difference of π across the (single) DC SQUID, which is its point of minimum (and negative) Josephson inductance. The result is that unlike the JPSQ, the persistent current of the flux qubit is determined by a balance between the Josephson inductance of the DC SQUID and that of the two loop junctions in series, which are of similar magnitude. Only near the point β = 1/ √ 2, where the potential minima correspond to ±π/2 across the DC SQUID (and where its Josephson inductance diverges) does the persistent current approachĨ Ca (φ ∆ ).

B. Analysis of fluxon tunneling in the JPSQ
We now discuss tunneling between the JPSQ's persistent current states, denoted here by | + z and | − z , using the Euclidean path-integral representation for the transition amplitude between them: . For the tunable flux qubit, this quantity is related to the well-known parameter α according to: α = 1/2β, such that the excluded region β < 0.5 corresponds to α > 1.

−z|e
Here, D[ φ(τ )] indicates a functional integral over all paths φ(τ ) = (φ I (τ ), φ l (τ )) connecting the classical minima of the two potential wells, and S E [ φ(τ )] is the corresponding Euclidean action associated with each such path. Anticipating the validity of the dilute instantongas approximation [46], we focus on the subset of paths satisfying these boundary conditions which correspond to a single instanton (i.e., those that pass over the barrier only once): φ (1) (τ ). Anticipating a semiclassical, stationary-phase approximation, we further subdivide these paths into two groups φ R (τ ), corresponding to those suitably close to one or the other of the two classical, extremal paths φ cl L (τ ) and φ cl R (τ ): Evaluating the functional integrals in eq. 14 requires finding the classical extremal paths φ cl L (τ ) and φ cl R (τ ) which satisfy the instanton boundary conditions: where T ∈ {L, R}. Before tackling this, however, we note that there is already something we can say purely based on the boundary conditions of eq. 15. The Euclidean action associated with each of the two classical paths is composed of two terms: where the first is due to the tunneling dynamics, which we will discuss shortly, and the second to interaction with the bias source polarizing the island with charge Q b . Due to the Josephson symmetry of the potential with respect to φ I , we can make the following statement about the latter term in the action: where these contributions to the Euclidean action are imaginary because the voltage V = d Φ/dt → −i V under the Wick rotation to imaginary time t → iτ . The integration path subscripted I in eq. 17 indicates a single, closed path encircling the island, which corresponds to a translation by exactly one period of the Josephson symmetry (note that this symmetry is not broken when δφ x = 0). Therefore, as shown in eq. 18 and expected from our intuition for the Aharonov-Casher effect, the relative phase between transition amplitudes corresponding to the two classical tunneling pathways is given simply the dimensionless polarization charge applied to the island q b . This relative phase shift can be viewed as a geometric effect [47], associated with the area enclosed by the path I in (Φ, Q) phase space [39,48], [49]. We now return to the problem of calculating the first term in eq. 16, starting by writing down the two saddlepoint energy barriers separating the persistent-current states, and traversed by the two tunneling paths T ∈ {L, R}: where ± here and below refer to these two paths. From eqs. 19 we see that whenδφ x = 0, one of the barriers is lowered and the other is raised, as required to realize the situation pictured in fig. 3(d). Now, although the aforementioned extremal paths contain these points, solving exactly for these paths in 2+1D (already neglect-ing motion along the φ p direction as discussed above) requires classically integrating the equations of motion, taking into account the fact that the inverse capacitance matrix has nondegenerate eigenvalues (that is, the fictitious phase particle has an anisotropic "mass"). Since we are seeking here to obtain simple, analytic expressions useful for understanding the qualitative physics of this circuit, we use a simpler approach: for each group of paths φ (1) T (τ ) (T ∈ {L, R}), we confine the functional integral to linear paths in (φ l , φ I ) space connecting both potential minima and the intervening saddle point, as illustrated in fig. 4(b)-(e). To facilitate this, we rotate to a set of coordinates (φ T , φ ⊥ T ) for each path such that the approximate tunneling dynamics occurs only along the φ T direction (and we can neglect the contributions from motion along φ ⊥ T , by approximating it as separable and harmonic): where we have expanded to leading order in δφ x . We have also chosen the overall scaling of φ T so that the sum of the lengths of the two 1D tunneling paths is 2π, and the resulting problem can be viewed as occurring on a circle, as illustrated in fig. 4(f). Using eqs. 20 and 9, we can evaluate the resulting 1D potentials along the two paths U T (φ T ) ≡ U J (φ T , φ ⊥ T = 0), examples of which are shown in figs. 4(f). Along these paths, the potential minima are located at: and the effective Josephson inductances L −1 T m ≡ L −1 m + δL −1 m at these minima are given by: We can make the corresponding transformation of the circuit's inverse capacitance matrix, starting from the {q I , q l , q p } representation, where it is given by: with the definitions: C tot I ≡ C I + 2 (2C Ja ||C J ), C tot l ≡ C Ja + C J /2, and C tot p ≡ 2C J + (C I ||4C Ja ). From this, we obtain the inverse capacitance along the q T direction C −1 T ≡ C −1 + δC −1 [50]: where we have defined the ratio: r C ≡ C I /4C Ja . In order to find a simple analytic solution to the equations of motion in the two double-well potentials, we approximate them using the following sixth-order polynomial form: By construction, eqs. 25 have the same barrier height E bT and potential minima at φ T = ±φ T m as the exact potential derived from eqs. 9 and 20. We can match the Josephson inductance at the local minima φ T m as well by choosing the (small) sixth order correction coefficient k 6T to be: Figure 4(g) illustrates these potentials for the left and right tunneling paths, where solid lines are the exact results, and dashed lines the approximation described by eq. 26 (note that they are nearly indistinguishable). Using this form for the potential, we can readily find solutions for the imaginary-time equation of motion at zero energy, given by: with the boundary conditions φ T (−∞) = −φ T m and φ T (∞) = φ T m (equivalent to real-time dynamics in the inverted potential −U T (φ T ) at zero energy). The simple polynomial form of eq. 25 allows eq. 27 to be integrated to obtain: where τ 0 is an arbitrary position in imaginary time (known as the collective coordinate of the instanton), and is the frequency of small oscillations about the potential minima. Using this solution, we can evaluate the corresponding Euclidean action analytically by expanding in powers of the small parameter k 6T and integrating: where Z T = C −1 T /L −1 mT . Using eqs. 21, 22, 24, and 26, we can now obtain: where the two terms in the last line correspond to the δφ x -dependence of the barrier height and oscillation frequency, respectively, Ω ≡ 1/ √ L m C is the average singlewell oscillation frequency, and we have neglected the small contribution to δs from the δφ x -dependence of k 6T , by defining: To get a more intuitive picture of the important parameter dependencies, we consider the large-β (small-θ) limit, obtaining to leading order in 1/β: where we have defined the bare junction plasma frequency ω J ≡ 1/ √ L Ja C Ja and the dimensionless DC SQUID admittance: y Ja ≡ R Q /Z Ja , with Z Ja ≡ L Ja /2C Ja , and R Q = h/4e 2 is the superconducting resistance quantum. The parameter y Ja describes the importance of quantum phase fluctuations across each DC SQUID, with the large-y Ja limit corresponding to semiclassical behavior (small phase fluctuations), and the tunneling action proportional to y Ja . This quantity, in combination with β and r C , are the fundamental dimensionless parameters of the circuit in this simplified model. In order to perform the path integrals in eq. 14, we generalize well-known results for the quartic potential [46]. In computing the usual fluctuation determinant describing Gaussian fluctuations about each stationary path, we treat the effect of the small sixth-order term in each potential using first-order perturbation theory. We account for the contributions of both stationary paths by viewing our quasi-1D problem as a particle on a circle with two potential minima [c.f., fig. 4(f)]. We obtain, in the dilute instanton gas approximation, the following result for the Euclidean transition amplitude: where we have defined the fluxon tunnel splitting frequency: Combining eqs. 11 and 32, we obtain the following effective Hamiltonian for the two lowest-energy states (up to an overall energy offset, and a static rotation around z), valid for small δφ z , δφ x , and arbitrary q b : As expected, when Q b = e (q b = π) and δφ z = δφ x = 0 in eq. 34, we haveĤ = 0 due to destructive Aharonov-Casher interference between the two tunneling paths in eq. 14 [51]; this corresponds to the emulated zero field point around which we wish to operate.

C. Dipole moments of the JPSQ
Focusing on the regime near this point, we formally rewrite the Hamiltonian of eq. 34 in terms of a Zeeman-like interaction between an effective vector dipole moment operatorˆ µ, and an effective field F, as follows: where we have defined δQ y ≡ Q b − e, andˆ µ is given at zero field by:ˆ µ ≡ (I xσx , V yσy , I zσz ) These dipole moments govern the strength with which the JPSQ couples (in the computational space) to external fields, including classical fields used to manipulate it, fields from other qubits that are used to engineer entangling interactions, and fields from its noise environment that are responsible for decoherence. This is, of course, a general feature of nearly any qubit system, that the same interactions with external fields that provide a mechanism for using the qubit also open the door to decoherence processes. Figure 6 shows a comparison between the predictions of eqs. 34-36 and full numerical simulations of the circuit, performed using a generalization of the methods described in refs. 37 and 40 [38]. The abcissa for the plots is the parameter β(φ ∆ ), which describes the ratio between the Josephson inductance of the DC SQUIDs and that of the large junctions. Panels (a)-(d) are for y J = 25 and panels (e)-(h) are for y J = 15. The different colors in each plot indicate different values for the dimensionless island shunt capacitance r C . The leftmost column, panels (a) and (e), shows the energy splitting Ω ge /2π in a more conventional flux-qubit-like regime, where δφ z = δφ x = Q b = 0 (corresponding to zero z and x fields, and a maximal y field). The remaining three columns show the three components of the dipole moment, I x , V y , and I z , near the qubit's emulated zerofield point. Both the energy splittings and the dipole moments decrease strongly with increasing r C , a trend that can be understood from eqs. 24 and 31: in the β 1(θ 1) regime of most interest here, the effective inverse capacitance C −1 T that acts as a "mass" for fluxon tunneling is mostly controlled by C I . Note that this is somewhat different from an ordinary flux qubit, where the corresponding tunneling "mass" is controlled largely by the capacitance across the (single) small junction (or DC SQUID in the case of a two-loop qubit [26,27]).
We make two general remarks about the agreement between our analytic results and the full simulations shown in fig. 6. First, the agreement is much better for the larger value of y Ja (top row) than for the smaller (bottom row). The agreement is also better for larger values of r C . Both of these trends are to be expected, since both of these parameters control the validity of the semiclassical (stationary phase) approximation used to derive eq. 34; that is, larger y Ja and r C both result in smaller quantum phase fluctuations. Second, in many cases the agreement is substantially worse for small β. This is also to be expected, since as β is decreased, the relative importance of quantum fluctuations along the φ p direction increases, as can be deduced from eq. 9 by calculating the effective impedances for small oscillations about the potential minima along the three mode directions.

D. Numerical simulation of realistic JPSQ circuits
Although the circuit of fig. 3(b) and the corresponding results of eqs. 34-36 capture the most important qualitative features of the JPSQ, full numerical simulation and a more realistic circuit description are helpful to fill in additional important details. Figure 7 shows a JPSQ circuit which includes finite geometric loop inductances in the DC SQUIDs, and fig. 8 shows the low-lying energy levels obtained from numerical simulation [38,52] of this circuit. The energies are plotted as a function of δΦ z , δΦ x , and Q b , for parameters that correspond to those of fig. 6, with β = 15, r C = 0.1. In contrast to the analytic treatment in the previous section, numerical simulation allows us to look in detail at the higher energy levels of the circuit outside the computational subspace, which is important for understanding when these higher levels can safely be neglected. As shown in figs. 8(b) and (d), at energies greater than ∼ E b (the height of the fluxon tunnel barrier, labelled in fig. 8 with a vertical arrow) above the ground state, the Q b -dependence increasingly looks like that of the simple charging Hamiltonian for the circuit's island, given by (up to a constant energy offset): where E CI ≡ e 2 /2C tot I is the island charging energy and C tot I = 4C Ja + C I for the circuit of fig. 7. Referring back to fig. 4(a), we can immediately get an intuitive picture for the energy of the lowest excited states above the computational space: as the energy increases above the height of the fluxon tunnel barriers, the double-well Colored arrows indicate flux bias offsets that are added to the gauge-invariant phase differences across the circuit's loop inductors. The circuit Hamiltonian is described in terms of one Josephson mode (described in a charge basis) and six oscillator modes (described in an oscillator eigenstate basis) [38]. Of the latter, three are linear oscillators and three have potentials that contain both Josephson and linear inductive terms. Note that the capacitors C0 are required to make the capacitance matrix singular, but their numerical values are kept small enough to have negligible influence on the results. potential in each unit cell along the φ l direction becomes increasingly unimportant, and the states look more and more like plane waves (charge states) along the φ I direction, governed by eq. 37. This allows us to identify the next two higher excited states near Q b = e approximately with the two island charge states Q I /2e = {−1, 2}, whose energy above the qubit levels is ≈ 8E CI . These levels will act as an upper bound on the energy scale over which we can treat the JPSQ as a two-level quantum system (similar to the so-called "anharmonicity," which is used to specify the importance of the second excited state in the context of transmon and flux qubits).
Another piece of information evident in fig. 8 is the importance of the island parity, indicated by black and blue lines for even and odd island parity states, respectively. Focusing on the region around δφ z = 0, Q b = e, we see that the odd-parity ground state is actually lower in energy than the two lowest even-parity states which form the computational subspace. This means that these even parity states can decay into the lower-energy odd parity ground state near this point, if a quasiparticle tunnels onto the island inelastically [53]. Hence, the so-called "parity lifetime" of the island, a quantity well-known in the literature of superconducting and semiconducting qubits [53][54][55], is crucially important in assessing the potential coherence of the JPSQ. Using the fact that the quasiparticle tunneling operator for a JJ which connects its even and odd-parity charge subspaces is proportional to sin (φ/2), whereφ is its gauge-invariant phase difference operator [53,54], one can readily verify that these inelastic, island-parity-changing quasiparticle tunneling We have also taken L∆ = 50pH, and C0 = 0.1 fF. Note that C0 must be nonzero for the capacitance matrix to be nonsingular, but the simulated energies depend negligibly on its numerical value as long at it does not become comparable to CJa. The two horizontal axes of (a) and (c) are the flux biases corresponding to the emulated z and x fields, and the zero field point is indicated by a filled black circle. Black (blue) solid lines are the eigenergies for even (odd) charge parity of the island. Panels (b) and (d) show the island charge dependence of the energy levels at this zero field point, where the vertical magenta arrows indicate how the levels between each pair of plots are connected: the island charge is varied from Q b =0.5 to Q b =0 starting from the zero field points in (a) and (c). A wider energy range is shown in (b) and (d) to illustrate the form of the island charge dependence: for energies greater than the top of the fluxon tunnel barrier (indicated by E b ), the level structure increasingly matches that of the simple island charging Hamiltonian of eq. 37. Red dashed lines in (b) and (d) show the eigenenergies of eq. 37, where an overall energy offset has been added such that they match the simulation at high energy.
events are strongly allowed in the JPSQ for all bias conditions [56]. As a result of this, both the ground and excited states in the (even parity) computational subspace will have decay rates 1/T 1 of the order of the quasiparticle current spectral density S qp (ω) defined in ref. 53. For Aluminum junctions, typical quasiparticle densities, and parameters in the range considered here, these lifetimes could be as short as the ∼ µs range. Fortunately, there exists a method for almost completely suppressing these processes, which has been used in both superconducting and semiconducting circuits to increase island parity lifetimes to the millisecond range and beyond [55]. Referring to the circuit of fig. 7, one need only use a higher-gap superconducting material for the island as compared to the rest of the circuit, such that quasiparticles occupying the island see a higher potential energy. Once this potential barrier becomes substantially larger that the sum of the thermal energy and the maxmimum emulated Zeeman energy, the circuit is effectively protected from noise-induced parity-changing transitions.

E. Discussion of JPSQ coherence and comparison with existing superconducting qubits
Assuming that the parity switching discussed above can be circumvented as in previous works [55], the dominant intrinsic sources of noise in these circuits will be the same as for other superconducting qubits. These can be described in terms of charge and flux noise, both of which exhibit high-frequency and "1/f-like" low-frequency components that have been observed under a variety of conditions [28,40,57,58,60,61]. As described above, the sensitivity of the JPSQ to charge and flux noise can be described simply in terms of its dipole moments [c.f., eqs. 36]. Table I shows these moments calculated for three sets of JPSQ parameters, labelled cases A,B, and C, corresponding to circuit designs with increasing magnetic dipole moments (with I x ∼ I z in each case) spanning a range from ∼40 nA to ∼250 nA. Because the JPSQ is engineered to emulate a vector spin-1/2, these moments are, by design, approximately independent of field around the emulated zero-field point, allowing decoherence processes to be viewed in the same simple manner used to describe the response of a spin-1/2 to field noise. In the presence of a nonzero externally-applied offset field, noise fluctu-  fig. 3 and fig. 7. From the latter, we derive the two coherence metrics Γ g φe and T1, which describe the longitudinal (dephasing) and transverse (population transfer) components of the decoherence, respectively. The first of these, Γ g φe , is defined in refs. [57,58], and characterizes the effect low-frequency noise; it is the rate coefficient in the Gaussian (for 1/f noise) expression for the envelope decay of the system's response to a spin-echo pulse sequence. Although pulsed operation is not the focus of this work, we use this metric because it can be specified without requiring artificial cutoff parameters for the noise spectrum [57]. This complication is a result of the singular character of the noise at low frequencies, which also produces the non-exponential decay of coherence. Because of this non-exponential character, (Γ g φe ) −1 cannot meaningfully be viewed as, or compared to, the usual "dephasing time" T2. The second metric is T1, the lifetime of the qubit excited state. Here, we tabulate the lifetime that would result from the expected noise power spectral density at 5 GHz (resulting from an applied perpendicular field): SQ(h×5 GHz) ∼ (1.1 × 10 −8 e) 2 /Hz, and SΦ(h×5 GHz) ∼ (4.3 × 10 −11 Φ0) 2 /Hz. These noise levels are derived from the results of ref. [40], which were themselves obtained using data from a large number of flux qubits of varying design and over a range of bias conditions.

Qubit type
Two-loop flux [27] Fluxmon [28] D-Wave flux [20] Transmon Flux qubits' I x and V y go exponentially to zero as F x → 0, while for a transmon I z goes linearly to zero near its maximum energy splitting (often called a "sweet spot"). Here, we tabulate values for bias far from these points. † Since for flux qubits I x → 0 as F x → 0, noise in F x only produces T1 processes when both F x and F z are nonzero, and the resulting rate depends in detail on both of these quantities. ‡ The very small static charge dispersion of these qubits makes them highly insensitive to dephasing from charge noise. † † Viewed as a spin, the transmon can be described as experiencing a large offset field, which points purely along z if its two junctions are symmetric. Therefore, flux noise in F z can only produce nonzero transverse T1 processes if this symmetry is broken.  [43] is also a persistent current qubit; however, we have not included it in the present comparison because its persistent current is, by design, too small to be used for the direct spin emulation discussed here). The final column is for the transmon qubit widely used in gate-model applications, which is included as a point of reference.
ations are naturally divided into their longitudinal and transverse components, relative to the axis of that field. Longitudinal noise fields cause the spin's Larmor precession frequency (Zeeman energy) to fluctuate, resulting in so-called "dephasing" (T 2 processes) whose magnitude depends mostly on the low-frequency content of the noise power spectrum. These processes are represented by the quantity Γ g φe shown in the table for each case, and for each moment. Transverse noise fields cause the spin's precession axis to rotate, resulting in population transfer (T 1 processes) in the computational basis that depends mostly on the noise power spectral density at the Larmor frequency, described by the quantity T 1 in the table.
For comparison, table II provides typical values for existing, state-of-the art superconducting qubits. Its first three columns show values for three demonstrated examples of tunable flux qubits, to which the JPSQ is most sensibly compared, and its final column the well-known transmon qubit used for nearly all gate model applications [59]. Broadly speaking, tables I and II exemplify the fact that unlike the transmon qubit, whose simplicity results in very little design freedom, the dipole mo-ments of JPSQs and flux qubits can vary widely, according to the needs of the designer: tunable flux qubits have been used with I z values ranging from the ∼ 50nA in recent capacitively-shunted flux qubits [40] (with coherence times as high as ∼ 50µs), to ∼ 3µA in the machines from D-Wave systems [8,20,21] (with coherence times in the range of tens of nanoseconds). Correspondingly, unlike the transmon qubit, whose coherence in the ideal case is largely set by fundamental material properties and physical geometry, the coherence of flux qubits and JPSQs also depends very strongly on the specific design requirements set by the system in which it is used [62]. In spite of this fundamental difference, one simple coherence comparison that can still readily be made between the JPSQ and both flux and transmon qubits is in their sensitivity to high-frequency noise: the data for JPSQ cases A and B in table I clearly show that JPSQs can readily be designed with comparable or longer T 1 times than the flux qubits and transmon listed in table II [63].
In order to compare the dephasing in these circuits, we first note the following: both flux and transmon qubits have a so-called "sweet spot," a bias point where their energy splitting becomes linearly insensitive to both lowfrequency flux and charge noise (the latter is due to a large ratio of Josephson to charging energy, and is not dependent on bias). At such a bias point, we have, by definition: e|ˆ µ|e − g|ˆ µ|g = 0; that is, at a sweet spot the qubit's static dipole moment is zero. This is obviously a useful property in some cases, as it allows the qubit to be decoupled from noise [57,58]; however, it is incompatible with emulation of a vector spin-1/2 (note that this is simply a more general restatement of the conclusion of section I). Therefore, the JPSQ, or any circuit engineered to emulate a vector spin-1/2, must by its very design be open to additional dephasing channels as compared to flux and transmon qubits, which cannot. For the JPSQ, this additional dephasing manifests itself it two ways: First, compared to a conventional flux qubit, the JPSQ experiences dephasing due to noise in F x at all values of ∆E x , whereas the flux qubit has a sweet spot when ∆E x is at its minimum value, and is only subject to dephasing from F x noise away from this point. As can be seen from the tables, the values for I x are generally comparable to or smaller than I z ; so, while this additional sensitivity will result in larger total dephasing rate, the difference will be less than a factor of two.
Secondly, unlike both flux qubits and transmons, the JPSQ has a static V y , making it sensitive to lowfrequency charge noise, which is in general a more serious concern. Obviously, one would like therefore to minimize this sensitivity by keeping V y as small as possible. However, from eqs. 36 and 31 we can see that the value of V y is closely tied to that of I x . In fact, the exponential dependence of I x on the tunneling action S 0 implies, for a given I x (assumed to be set by external system requirements), that β tan(φ ∆ /2) is the only accessible parameter on which the resulting V y depends more strongly than logarithmically (for β 1). This is evident in table I, where the ratio V y /I x (with units of impedance) is nearly identical in all three cases, and leads to the same design conclusion as fig. 5: that β should be made as large as possible [64]. Examining the corresponding JPSQ dephasing rates due to low-frequency charge noise in table I, we see correspondingly that they are in all three cases about twice as large as those due to flux noise coupling to I x . So, although this additional charge noise dephasing is unavoidable for the JPSQ, it will at worst increase the total dephasing rate only by this modest factor.

III. EXAMPLES OF MULTI-JPSQ CIRCUITS
We now give some examples of how spins emulated using the JPSQ circuit can be coupled together in ways that have not previously been possible with engineered quantum devices. Figure 9 shows our first example, in which two of the JPSQ circuits detailed in fig. 7 are coupled to each other via a pair of tunable RF SQUID flux qubit couplers [21,37]. One of these is magnetically coupled to the I z dipole moments of the qubits, and the other to their I x moments. Panels (b)and (c) show the lowest energy levels of this circuit, obtained by full numerical simulation [38,52]. For panel (b), the zz coupler is turned on (Φ Cz = Φ 0 ) and the xx coupler is off . Focusing on the lowest four energy levels, which act as the two-spin computational subspace, we see that strong two-qubit zz and xx coupling are both possible with this circuit just by adjusting the flux controls of the two couplers. Furthermore, the two types of coupling are qualitatively equivalent, as evidenced by the fact that the two panels, one with only xx coupling turned on and the other with only zz, look nearly identical, except that the roles of x and z are permuted [compare to the strong non-equivalence seen for a two-loop flux qubit shown in fig. 1(h)]. This equivalence is a result of the JPSQ's ability to simultaneously emulate a true rotational symmetry at its zero-effective-field point while maintaining a strong, vector dipole moment.
The circuits we have discussed so far exploited only the x and z components of the JPSQ dipole, which are both magnetic. Figure 10 shows a circuit which makes use of all three of its vector components. This circuit contains four coupled JPSQs (each like the one shown in fig. 7) which together realize a single logical spin with passive quantum error suppression, based on a distance-2 Bacon-Shor code [3,4]. The JPSQs are connected by strong, pairwise couplings, with magnetic zz couplings between the qubit pairs (1,3) and (2,4), and electric yy couplings between pairs (1,2) and (3,4). These interactions correspond to the two-qubit check operators for the code, and together they produce a 2-dimensional logical computational subspace whose effective logical spin operators are products of two single-qubit physical operators, as shown in fig. 10(a). Because the logical spin  fig. 3(d)], resulting in the desired xx coupling. Panels (b) and (c) show the resulting energy levels when one of the couplers is turned on, and the other off, producing pure zz coupling and pure xx coupling, respectively. These two bias configurations show an evident symmetry under permutation of the labels x and z (Note that for clarity these plots show only the energies on the boundaries (δΦ x , 0) and (0, δΦ z ) rather than full energy surfaces). This indicates that a true rotational symmetry (in this case around y) can be engineered, something that is not possible with any present-day superconducting qubit circuits.
operators are 2-local, the two logical states in the ideal case are protected against all single-qubit noise on the four constituent physical qubits; or, more precisely, any single qubit noise process should couple the two logical states only to higher-energy levels, separated by an energy "barrier" whose height is of the order of the strength of the strong pairwise interactions. Therefore, singlequbit noise processes in the four constituent qubits in the ideal case must supply at least this amount of energy to affect the encoded logical spin state.
The circuit of fig. 10(b) also contains two additional two-loop RF SQUID flux qubit couplers (each like the couplers shown in fig. 9), which are used to implement a zz interaction between qubits 1 and 3, and an xx interaction between qubits 1 and 2. As illustrated in fig. 10, these two interactions correspond to the logical x and z operators, respectively. Adjusting the strength of these two-qubit couplings controls the effective field seen by the logical spin. Figure 11(a) shows the resulting dependence of the simulated [38,52] energy levels (relative to the ground state energy) on these two logical field controls. Around zero field, the resulting energy barrier separating the logical states from the first excited man-ifold (corresponding to violation of one of the penalty interactions) is ∼ h × 2.1 GHz∼ 5.3k B T , corresponding to a Boltzmann factor of 5 × 10 −3 . This number can be viewed as the relative thermal occupation of environmental photons at the low frequencies which separate the two logical states, and those that connect them both to the next higher set of states.
We now wish to evaluate the sensitivity of this protected qubit to decoherence arising from local, physical flux and charge noise. We define the c-number dipole moment D ij [Ô] associated with a dipole operatorÔ and a two-dimensional subspace {|i , |j } as: where W [O ss ] is the numerical range of the 2 × 2 matrix s|Ô|s , and s, s ∈ {i, j}. We make the simplifying assumption that each circuit node experiences independent charge noise of the same magnitude, and each geometric inductor experiences independent flux noise of the same magnitude. We can then define the following effective to- tal electric and magnetic dipole moments to which these common noise levels can be said to couple in the twodimensional subspace {i, j}: and which can be compared to the corresponding dipole moments of physical qubits listed in tables I and II. Figure 11(b) and (c) show the results for these dipole moments in the logical computational subspace (i, j = 1, 2) derived from our simulations of the circuit of fig. 10(b). The electric dipole moment is in the ∼nV range, thousands of times smaller than typical superconducting qubits, suggesting that its T 1 due to charge noise would be millions of times longer [65]. The magnetic dipole mo-ments shown in fig. 11(c) are in the ∼nA range, set completely by the fact that we have intentionally engineered logical field controls via the two additional couplers [c.f., the shaded red and blue subcircuits in fig. 10(b)] and in so doing to we have effectively "poked a hole" in the error suppression. The ∼nA scale of these magnetic dipoles results simply from choosing to produce an ∼ h × 1 GHz logical Zeeman splitting over the maximum coupler control flux tuning range of Φ 0 /2. Much smaller magnetic dipole moments could trivially be achieved simply by reducing the accessible tuning range of the Zeeman energy; in the extreme case where we remove the logical field control couplers entirely (reducing this tuning range to zero), our simulations show residual magnetic dipole moments less than 1 pA. The key question here, in assessing the potential coherence of such a logical spin, is the manner in which it would be used. In the context of quantum annealing, one would like to retain a wide tuning range for the Zeeman energy, and the ∼ nA dipole moments shown in fig. 10(e) are therefore likely the smallest possible, if a single loop is used for logical field control (restricting the full flux tuning range to ≤ Φ 0 /2). However, this is already ∼100-10,000 times smaller than the magnetic dipoles of existing flux qubits [c.f., table II].
In addition to the spurious couplings of noise inside the logical computational space, we must also consider processes in which the system is excited out of this space, requiring the environment to supply a photon at an energy equal to that of the energy barrier [c.f., ∆E L in fig. 11(a)]. Although these processes are suppressed by the low temperature, they are also "allowed" transitions for single photons coupling locally to the circuit, as opposed to the couplings inside the computational space just discussed, which only occur due to spurious non-idealities in the circuit realization of the Hamiltonian of fig. 10(a). Figures 11(d) and (e) show the effective total rms dipole moments for these transitions from the two computational states to all of the states in the lowest excited manifold. As expected, these transitions are much stronger. We can readily estimate the resulting lifetimes of the logical states due to these transitions using the following flux and charge noise amplitudes at the frequency corresponding to the barrier height ∆E L /h ∼ 2.1 GHz, derived from the work of ref. 40: S Q (2.1 GHz) ∼ (7.1 × 10 −9 e) 2 /Hz, and S Φ (2.1 GHz) ∼ (6.4 × 10 −11 Φ 0 ) 2 /Hz. Assuming a thermal environment at T =20 mK so that the rates of absorption and emission processes at this frequency are related by the Boltzmann factor, we find in the region near zero logical field, a lifetime of ∼ 10 seconds due to charge noise, and 1.5-6 milliseconds due to flux noise. The latter, in fact, appears to be the coherence-limiting process for the parameters we have chosen, though it can likely be further improved by additional optimization of circuit parameters beyond that carried out here, with the specific goal of reducing the influence of these high-frequency magnetic transitions.
To employ such logical spins in a quantum annealing machine requires static, pairwise, logical Ising interac- f., eq. 39] of the logical states, with respect to physical noise fields, as a function of the logical z and x control fluxes (note that decoherence rates will in general scale with the square of these quantities). Panels (d) and (e) give the total rms electric and magnetic dipole moments, respectively, for excitation out of the logical space (from logical state |0 shown in red, and from logical state |1 in blue) into the first excited manifold of states |b shown in (a), separated by the energy barrier shown with a magenta arrow.
tions between them. These would correspond to physical four-qubit static interactions, which have not yet been demonstrated experimentally. However, due to their relevance for applications such as adiabatic topological quantum computing [6] and adiabatic quantum chemistry simulation [10], there are already several concrete proposals for realizing them [69]. A protected qubit like that shown in fig. 10 could also be used in gate model applications. In the simplest case, the logical field control could be used to implement single qubit rotations (via Larmor precession around the emulated vector magnetic field), though two-qubit gates would still require four-physical-qubit interactions in some form. However, since only pulsed operations are required in a gate model context, they would not need to be static nor very strong. One possibility would be to use the techniques described in ref. 39, which are based on state-dependent geometric phases accrued by the system when coupled to a resonator and driven, and which are readily scalable to multi-qubit entangling interactions. In this mode of operation, the logical field controls would no longer be necessary (reducing the magnetic dipole moment for physical noise to the ∼pA level, as mentioned above), and single and two-logical-qubit gates would be realized via selectively modulating the couplings between either two or four physical JPSQs, respectively, and a common resonator.

IV. TWO-ISLAND JPSQ CIRCUIT FOR EMULATION OF A 3D MAGNETIC MOMENT
So far, we have discussed a JPSQ circuit that can be said to emulate a quantum spin-1/2, insofar as it has three physical operators that can be engineered to obey the canonical commutation relations in the computational subspace of the lowest two energy levels, while FIG. 12. Emulating a Heisenberg quantum spin-1/2 with a JPSQ circuit. Panel(a) shows a circuit having two islands, and therefore three fluxon tunneling paths. The third tunneling path is realized with a single, fixed junction, while the other two are flux-tunable DC SQUIDs. The latter are biased with static flux offsets such that their fluxon tunneling amplitudes have flux sensitivities of equal magnitude. Panel (b) shows how by choosing the island charge offsets to be one third of a Cooper pair, the three tunneling amplitudes can be chosen to form an equilateral triangle in the complex plane, resulting in complete cancellation of tunneling. Panels (c) and (d) show how small changes to the two DC SQUID fluxes can then correspond to two orthogonal magnetic moments, via the common and differential modes of the two fluxes.
higher levels are kept relatively far away. However, only two of these operators are magnetic, and the remaining one is electric. In the case of fig. 10, only strong, fixed electric interactions are needed, which can be realized with the simple capacitive couplings between islands shown in fig. 10(b). However, in some of the applications mentioned in the introduction, one needs tunability of the couplings between all three components of the spins' dipole moments. Although there are proposed methods for engineering tunable electric couplings between superconducting circuits [67], they tend to require the qubits to have sufficiently large electric dipole moments that charge noise is likely to become a major problem. Therefore, it would potentially be of great interest to realize a JPSQ circuit whose magnetic moment has three independent vector components that satisfy the spin-1/2 commutation relations, so that the well-established techniques for controllable magnetic coupling could be brought to bear to realize fully-controllable, anisotropic Heisenberg interactions.
Such a circuit is shown in fig. 12. Here, we have added an additional fluxon tunneling path by including a third Josephson junction in the loop, and we bias the resulting two islands using separately-controllable voltages. As illustrated in fig. 12, if the polarization charges on these two islands are both set to one third of a Cooper pair, and the three tunneling paths have the same amplitude, a completely destructive, three-path Aharonov-Casher interference again occurs, as in the two-path case discussed so far. Because of the 2π/3 relative phase shifts between paths, we need only adjust two of the three amplitudes to control two (emulated) orthogonal transverse field directions, which can be accomplished using differential and common mode bias fluxes coupled to the two DC SQUID loops [66].

V. CONCLUSION
In this work, we have proposed for the first time a superconducting qubit circuit capable of emulating a spin-1/2 quantum object with a true, static, vector dipole moment, whose components can be chosen, to a large extent, by design. We analyzed this circuit in detail, providing some qualitative intuition for its basic properties, and validated this with full numerical simulation of its quantum Hamiltonian. Broadly speaking, our results indicate that JPSQs with reasonable design parameters, when compared to existing flux and transmon qubits, can have a comparable or lower sensitivity to high frequency noise, but a somewhat higher sensitivity to low-frequency noise, with the latter almost entirely attributable to (and a necessary consequence of) their qualitatively more complex emulation capabilities. The most important open question, for the coherence of JPSQ, is whether its parity lifetime can be sufficiently increased using the bandgap engineering techniques that have been demonstrated so successfully in other experiments [55]. If so, it would pave the way for high-coherence emulation of quantum spin-1/2 systems with nearly arbitrary pairwise vector interactions.
This capability would then enable the experimental exploration of a number of potentially significant ideas that until now have remained out of reach of available physical qubit hardware. These include, for example: the use of strong, non-stoquastic driver Hamiltonians in quantum annealing [23] and quantum simulation [7,8]; Hamiltonians required for Hamiltonian and holonomic computing paradigms [4,9]; and engineered emulation of the full quantum Heisenberg model. If combined with one or more of the proposed schemes for realizing static [69] or pulsed [39] multiqubit interactions, an even wider range of possibilities would become accessible, including adiabatic topological quantum computation [6], adiabatic quantum chemistry [10], and quantum error suppression [3,4].