Universal Set of Gates for Microwave Dressed-State Quantum Computing

We propose a set of techniques that enable universal quantum computing to be carried out using dressed states. This applies in particular to the effort of realising quantum computation in trapped ions using long-wavelength radiation, where coupling enhancement is achieved by means of static magnetic-field gradient. We show how the presence of dressing fields enables the construction of robust single and multi-qubit gates despite the unavoidable presence of magnetic noise, an approach that can be generalised to provide shielding in any analogous quantum system that relies on the coupling of electronic degrees of freedom via bosonic modes.


I. INTRODUCTION
A promising new approach in the field of trapped-ion quantum information processing has been the introduction of microwave and radio wave sources. One particular technique has involved making use of a static magnetic field gradient imposed along the trap axis to enhance particle interaction [1]. Such magnetic-gradient-induced coupling makes possible both individual addressing in the long-wavelength regime and the coupling of particles' motional and spin states even in the presence of negligible conventional Lamb-Dicke parameter. Crucial building blocks of the scheme have been experimentally realised, notably sideband coupling [2] and elements of conditional quantum logic [3].
A technique developed as an alternative to this approach has made use of oscillating magnetic fields inherent to microwave radiation to resolve the issues of coupling strength and individual addressing [4]. Implementation of microwave-driven single and multi-qubit gates using this route has been reported [5].
The inevitable challenge for the static gradient approach is shielding the system against magnetic noise in the experimental setting, due to the usage of magnetic-sensitive states. Pulsed decoupling [6,7] provides one possible strategy. Alternatively, the usage of dressed states for encoding the logical qubit [8][9][10] offers a possible shielding technique. The dressed-state approach has previously found applications in resonator and nitrogen vacancy systems [11][12][13][14][15] in addition to novel quantum gate designs for trapped ions [16,17] using laser and laser-microwave addressing.
Notably, the dressed-state approach in the context of long-wavelength quantum computing with static magnetic gradients was explored by Timoney et al. [18], demonstrating experimentally its feasibility. Improvements in qubit coherence times by more than two orders of magnitude have been reported. This exciting development holds the promise of robust, long-wavelength quantum computation in a set-up that is experimentally viable and easily scalable. * gmikelso@ic.ac.uk Here, we address the next task of building a universal set of quantum gates for the microwave dressed-state approach in the static magnetic gradient set-up. Basic single-qubit operations for such a system have been realised by Timoney et al. [18] and also by Webster et al. [19] in a slightly modified arrangement.
We develop in detail the set-up employed in [18] and propose a set of quantum operations that jointly enable the execution of universal quantum computing. Firstly, we show how to realise arbitrary single-qubit rotations, proposing several alternative gate schemes. Secondly, following the well-known scheme of Mølmer and Sørensen [20,21], we develop a two-qubit entangling gate. We simulate the gates numerically to demonstrate their experimental viability and present analysis of the key noise sources. Finally, we comment on the possibilities for extending our scheme to the experimental set-up employed by Webster et al. [19].

II. PHYSICAL SYSTEM AND DEFINITIONS
Our scheme retains all the key elements of the original proposal by Timoney et al. [18] (also described in Webster et al. [19]), including initialisation, read-out and encoding of the logical qubit with the help of dressed states. The particular candidate for experimental implementation is trapped 171 Y b + ions [19,22], however, the gate derivations are presented for a generic magnetic-sensitive fourlevel system, depicted in Figure 1. States |−1 and |1 are the magnetic-sensitive levels, and the presence of static magnetic field generates their splitting in energy. The |−1 ↔ |1 transition is considered forbidden in line with the 171 Y b + case.
One creates dressed states by means of a partial STIRAP sequence using the microwave fields Ω +/− , which is halted in the middle, leaving the fields on at constant strength. Choosing appropriate field phases enables one to reach ei-arXiv:1410.6720v1 [quant-ph] 24 Oct 2014 FIG. 1. Four-level system for the realisation of the dressed state qubit, together with couplings in the microwave and radio wave domain (Ω +/− and Ωg respectively). Rabi frequencies are denoted by Ωi, detunings by δi and laser phases by θi, φi. Another possible coupling not shown is between |0 and |0 , which is described using Ωz, θz and δz. States |−1 and |1 are the magnetic-sensitive levels, and the presence of static magnetic field is assumed.
ther of the dressed states: Experimental creation of such states has been achieved using 171 Y b + ions with lifetimes in excess of 500ms [18,19]. Quantum operations are to be carried out using either {|D , |0 } or {|B , |0 } as the logical qubit. The four-state system is viewed in either case by considering the remaining pair of orthogonal states: {|B , |0 } and {|D , |0 }, respectively. We also define 'up' and 'down' as alternative basis states, which will be important in the discussion: For the D-qubit: For the B-qubit: During the halted STIRAP sequence, with the dressing fields constant at Ω +/− = Ω, it is found that |u and |d diagonalise the Hamiltonian. Figure 2 plots the energy level diagram for the D-qubit case, showing how an energy gap is opened between the qubit space and the states |u and |d , an arrangement which could also be used for qutrit realization [23].
Interactions within the qubit space can be driven by introducing additional radio wave fields (Rabi frequency Ω g ). This arrangement provides the starting point for the single and multi-qubit gates presented in the paper.
It will be illustrated how single and multi-qubit gates can be realised in such a set-up, using, for the multi-qubit case,  (1) and (2). Analogous arrangement is found for the B-qubit. a magnetic field of constant gradient to strengthen the coupling between neighboring ions. In contrast to recent work, where second-order Zeeman shift is intrinsically used [19], we show how the simple first order shift is sufficient to construct a universal gate set. Further, we ease the experimental requirements by setting equal the phases and detunings of the radio wave fields: φ − = φ + , δ − = δ + (using distinct values for the hyperfine ground state of 171 Y b + would be possible, in principle, using polarisation). In other words, the radio wave couplings in Figure 1 would be created by a single field interacting with both |−1 ↔ |0 and |0 ↔ |1 pairs of levels simultaneously. In the case of the two-qubit gate (Section IV), interactions would be created by two radio frequency fields per qubit, which would each interact with both pairs of levels, thus generating four couplings per trapped particle.
Having demonstrated our scheme in detail, we discuss the case of non-linear Zeeman shift, considering modifications of our designs in light of the greater experimental facility (Section V).

III. SINGLE-QUBIT OPERATIONS
This section presents the techniques that enable universal single-qubit rotation to be executed on the dressed state qubit. We propose and describe two distinct gates (Sections III B, III D) as well as an adiabatic transfer technique (Section III C). Further, we mention two additional single-qubit gate designs, which are described in detail in the appendix.
Considering the eventual experimental implementation, within the set-up of an ion chain, addressing of individual qubits would be accomplished by separation in frequency space, with the help of static magnetic gradient [1]. This relies on gates coupling only such pairs of levels, where at least one state is magnetically sensitive, so that resonant frequencies vary along the trap axis. The gates proposed in this section do retain this property.
As the two key limiting factors to gate fidelity, we consider explicitly the noise in the ambient magnetic field and noise due to the instability of the microwave dressing frequencies Ω +/− . It will be shown how these effects can be overcome to reach gate fidelities in excess of 99% in numerical simulation. In order to maintain analytical tractability and illustrate precisely the role of the two sources of experimental noise, the single-qubit gates will be presented and analysed in the slightly simplified set-up with zero magnetic gradient present in the trap. Section III G provides justification for regarding the gradient a negligible effect for the single-qubit gates.

A. Hamiltonian and noise sources
We write down the single-particle Hamiltonian of the most general useful form ( is omitted throughout the paper). Figure 1 defines the phases, detunings and Rabi frequencies used. An extra possibility not drawn for clarity of presentation is the coupling between |0 and |0 , which is defined using Ω z , θ z , and δ z . Within the dipole approximation [24], one obtains the following expression: Moving to the interaction picture with respect to the timeindependent part (3) and performing the rotating wave approximation: Magnetic fluctuations are introduced by considering the additional term in the Hamiltonian, affecting the two magnetic-sensitive states: with µ(t) being a stochastic process of amplitude proportional to fluctuations in the ambient magnetic field.
Regarding the noise in Ω +/− , we approximate and define: where Ω is taken as constant and δ Ω is a second stochastic process. Since the radio frequency couplings will be generated, in the one-qubit case, by a single field, no analogous term is introduced for Ω g . The magnitude of the magnetic noise term µ can be quantified by its standard deviation SD µ . Section III F provides an estimate for this parameter, based on experimental measurements, of 2π · 88 Hz. In numerical simulation (Section III F), SD µ will be set constant to this value.
In contrast, the magnitude of δ Ω is modeled as being proportional to Ω. One can assume normally distributed noise in the strength of the microwave fields Ω +/− with standard deviation f Ω. Experimentally, Ω − and Ω + can be generated from the same microwave source that is multiplied by a radio frequency driving field. In that case, the noise in the microwave Rabi frequencies would be strongly correlated. However, under the extreme assumption of complete independence between Ω − and Ω + , the standard deviation of δ Ω would equal √ 2f Ω: In the experimental context, correlation between Ω − and Ω + would almost certainly reduce the value of SD δΩ significantly. However, (10) will be used in calculations and numerical simulation because of computational simplicity and for reasons of conservative estimation.

B. Basic σx/σy gates
Building on the work of Timoney et al. [18], it is shown how the σ y gate for the D-qubit and the σ x gate for the Bqubit can be realised by appropriate choice of field phases. Removing the |0 ↔ |0 coupling in (7) and choosing: one finds And setting: one obtains using the appropriate definitions of |u and |d (2). It is seen that the radio wave part (Rabi frequency Ω g ) in the above expressions yields the sought-after forms for the single-qubit quantum gates, while microwave dressing fields (frequency Ω) separate the energies of the remaining pair of basis states. The case of the D-qubit (12) has been plotted in Figure 2. The energy gap opened by the microwave fields plays a crucial role in shielding the qubit, particularly against the magnetic noise effects. Such a mechanism is common to all the gates presented in this paper.
Further examination reveals that the requirement to set equal the radio wave phases (φ − = φ + ) allows for no other σ i gate to be created using this route for either the B or the D-qubits. The scheme could be generalised to consider superpositions of states |B and |D , so that the logical qubit would now be represented by {|0 , cos γ|B + sin γ|D }. In such a case, a single σ γ gate in the xy plane of the Bloch sphere becomes feasible for each choice of γ. However, the technique allows for no second independent rotation to be achieved for the same definition of the logical qubit. Hence, complementary techniques will be required to realise universal single-qubit rotation.
Considering the D-qubit case and adding the two noise sources (8,9), expression (12) remains unaltered, but it needs to be complemented by the following term: Moving to the interaction picture with respect to the microwave and radio wave part (12), one finds that rotating phases of frequency (Ω ± Ω g )/ √ 2 are introduced to all terms in H n (15). Therefore, provided that the magnitudes of µ, δ Ω are much smaller than the rotation frequency, the terms can be deemed negligible within the rotating wave approximation.
The magnitude of H n (in the interaction picture) can be further estimated by adiabatic elimination [25], writing the time-propagation operator U (t) in orders of H n and looking for terms that grow linearly with t (secular terms). In the second order, one recovers corrections to the energies of |u and |d , in addition to terms in the qubit space: This amounts to an energy shift and a correction to the σ y gate couplings. In the third order, one finds population leakage terms out of the qubit space of magnitude: Minimisation of these unwanted terms can be accomplished by suppression through large denominator. The conditions for this can be summarised as:

C. Adiabatic transfer between |B and |D
The basic σ x and σ y gates can be linked for computational purposes by means of population transfer between |B and |D . This is achieved by adiabatic variation of the microwave phase in a set-up that leaves |0 decoupled.
Removing the |0 ↔ |0 coupling and the radio frequency fields in (7), one sets Ω +/− = Ω. This provides the timescale on which adiabacity would be maintained. One also sets to zero one of the microwave phases: θ − = 0. The transfer is based on slow variation of the other microwave phase θ + (t), such that the system is kept in the zero-eigenvalue state: Moving from |D to |B is achieved by varying θ + from 0 to π and moving from |B to |D is obtained by varying the opposite way. Given that |0 remains decoupled throughout, the following evolutions are enabled: The Berry's phase has been added in the expressions above, which can be calculated using standard formulae [26,27].
In the numerical simulations (Section III F), we vary the microwave phase continuously over a greater range, which yields an outcome state that is a straightforward linear extension of (20).
To analyse the effects of noise, one views the system in the adiabatic basis {|0 , |Ψ 0 , |u ad , |d ad }, where the noiseless Hamiltonian is diagonalised. The states {|0 , |Ψ 0 }, which represent the qubit space, lie at zero energy, while the latter two time-dependent orthogonal eigenstates are found to lie at energies ±Ω/ √ 2. This way, an energy gap is realised.
Applying the appropriate basis change to magnetic noise (8), and introducing effects due to microwave instability (9), one finds the following noise contribution: Moving to the interaction picture with respect to the noiseless Hamiltonian (Ω/ √ 2) · (|u ad u ad | − |d ad d ad |) will introduce rotations to all terms in H n , making them negligible within the rotating wave approximation for sufficiently large Ω.
Expanding the time-propagation operator in orders of H n (in the interaction picture) and looking for secular terms, one finds in the second order a term affecting the qubit space: The third order is found to contain leakage terms out of the qubit space of functional forms: µ 3 /Ω 2 , µ 2 δ Ω /Ω 2 , µδ 2 Ω /Ω 2 , δ 3 Ω /Ω 2 . Minimising these unwanted couplings requires: In contrast to the basic σ i gates, where the speed is governed by the radio frequency fields, the maximum speed of adiabatic transfer is governed by Ω and the requirement for the evolution to remain adiabatic.

D. Adiabatic σz gate
We construct a σ z gate based on adiabatic evolution and the Berry's phase. The gate idea follows the proposal by Duan et al. [28], although it is modified in important ways to suit the present set-up and improve speed and resilience.
The gate is illustrated for the case of the D-qubit, noting that analogous construction also exists for the B-qubit.
One removes the |0 ↔ |0 coupling in (7) and introduces adiabatic variables R 1 (t) and R 2 (t) as follows: Again, Ω fixes the adiabatic timescale for the gate. Substituting into the noiseless Hamiltonian (7) one obtains: It is seen that |D remains decoupled. The σ z gate will be created by inducing the Berry's phase in the |0 component, effecting the following evolution: This will be enabled by the zero-energy eigenstate of (25): To begin and end at state |0 , any adiabatic evolution of |Ψ 0 (t) in the {R 1 , R 2 } plane will need to begin and end on the line R 2 = π/2. The Berry's phase generated by any such trajectory can be calculated [26,27]: For the purpose of gate speed, it is desirable to find a path that yields the maximum phase while traversing the least distance. It is seen from (28) that moving along R 2 = π/2 will generate no extra phase. Figure 3 shows the path we propose, beginning at point A and ending at point D. Furthermore, the segment B → C is omitted, based on mathematical arguments.
One uses (28) to establish that no Berry's phase is generated along the segments A → B and C → D. In contrast, the phase generated along B → C is found to be Φ = R 1 (t) − x. This cancels exactly the time evolution of |Ψ 0 (27), so that along B → C the state follows as: displaying no time evolution. It is also seen that the Hamiltonian (25) effects no time evolution for |Ψ 0 along B → C, irrespective of the range x. These arguments allow one to cut out the segment B → C altogether, meaning that a trajectory of the same length can be traversed in the {R 1 , R 2 } plane to induce arbitrary phase for the |0 component. The total phase induced at the end of the path (see (26)) is found to be Φ = −x.
For the purpose of noise analysis, the Hamiltonian is diagonalised using the adiabatic basis {|D , |Ψ 0 , |u ad , |d ad }, where the latter two states are found to lie at energies ±Ω/ √ 2. Applying the basis change to the noise contributions, the following term is found: Line (30) yields a first-order noise term within the qubit space that is not correctable by the dressing field. After transforming H n to the interaction picture with respect to the noiseless Hamiltonian (Ω/ √ 2) · (|u ad u ad | − |d ad d ad |), the following extra contribution is found in the qubit space to second order: Moreover, leakage terms of forms δ Ω µ/Ω, δ 2 Ω /Ω are also recovered.
The dominant noise term is by far (30), which can be minimised by requiring good microwave stability (f 1), and by lowering Ω (and hence SD δΩ (10)). Considering the first and second order terms only would suggest that a choice of Ω as low as possible would minimise these lowestorder noise effects.
However, the third order analysis reveals terms that grow with reduced Ω. The following is found in the qubit space: In addition, leakage terms of the following form are found: The requirement to maintain negligible terms such as µ 3 /Ω 2 sets a lower limit on Ω, suggesting the existence of an optimal microwave dressing frequency. This is further confirmed in the numerical analysis.
Noise minimisation would therefore be achieved for: It will be shown in Section III F 3 how a value for Ω opt does indeed emerge numerically for different sets of simulation parameters. It is expected also within the experimental context that a value for Ω opt can be found beyond which a reduction in fidelity occurs. A further lower limit on Ω would be set by the desired gate speed and the adiabacity requirement.

E. Other σz gate designs
For completeness, other ways to realise the σ z gate are briefly described, taking the example of the D-qubit. Firstly, it is possible to construct the adiabatic σ z gate via two alternative routes. Section III D has demonstrated how a phase in |0 can be induced by employing couplings of the following form: |0 ↔ |B ↔ |0 (see (25)). Alternatively, one can induce the Berry's phase in |0 by employing couplings of form |0 ↔ |0 ↔ |B , in a set-up that uses Ω +/− and Ω z microwave fields. It is also possible to follow more closely the original proposal of Duan et al. [28], using the following couplings: |−1 ↔ |0 ↔ |1 . This arrangement would require microwave fields Ω +/− only and work by inducing a phase in |D . The disadvantages found for these alternative schemes include lower gate speed, less favourable noise effects, and the need to couple two magnetically insensitive levels. The adiabatic gate presented in Section III D is found to possess the most favourable overall qualities. However, it is also acknowledged that other functional forms for introducing the adiabatic variables {R 1 , R 2 } could be explored.
Secondly, it is also possible to use the effect of Stark shift [29] to create the σ z gate, a viable alternative to the adiabatic approach. We show two such designs in appendices A and B, the first of which relies on detuned |0 ↔ |0 coupling. It is shown how microwave dressing can be applied in such a case to shield the gate. The scheme would have the potential disadvantages of having to couple two magnetically insensitive levels, as well as having tighter experimental constraints on the parameters (A4). Likewise, we present in appendix B a radio wave Stark shift σ z gate that relies on {φ − = φ + , δ − = δ + }, which goes beyond the experimental limitations considered. The gate is added in light of extending the discussion to non-linear Zeeman regime (Section V), and is found to possess good shielding properties.

F. Numerical simulation
Experimental noise in the magnetic field and in the microwave and radio wave Rabi frequencies is modeled as the Ornstein-Uhlenbeck process, using formulae found in Gillespie et al. [30]. Two parameters need to be specified for each process: the relaxation time τ and the diffusion constant c.
For the simulation of magnetic noise, we obtain τ using the spectral density measurement provided by the experimental group of Wunderlich at Siegen [22]. Figure 4 plots the measurement in log-log coordinates, displaying an overall shape consistent with the Ornstein-Uhlenbeck model (smooth curve with a turning point) [31]. The turning point of the graph yields an estimate of τ = 0.1 ms. An estimate of c is obtained by calibrating to the lifetime of the |D state found in Timoney et al. [18]. The term µ(t) in (8) is thus found to have an estimated (fully relaxed) standard deviation of 2π · 88 Hz. Noise is added to the Rabi frequencies of the microwave and radio wave fields assuming the same relaxation time of 0.1 ms. An estimate for c is obtained by assuming noise standard deviation of 0.3%. In this paper's notation, we have used the estimate f = 0.003 for the microwave noise (see (9,10)). An analogous parameter f rw = 0.003 is introduced for the radio frequency field.
The fidelity of a quantum state ρ, with respect to a desired target or comparison state |Ψ c , is defined [32]: so that the probability of finding |Ψ c upon measurement is given by F 2 . There is a square root difference between this definition and the convention used in the paper by Mølmer and Sørensen [21].

Basic σx/σy gates
Results for the basic σ y gate are shown in Figure 5 top. Detailed simulation parameters are provided in the caption. The plot makes clear the efficiency of shielding against noise by means of the microwave dressing fields. At Ω = 2π · 40 kHz (the black curve), one obtains a reliable gate of almost vanishing noise contribution on the time scale considered. Averaging over 5 runs, the gate fidelity reaches F = 99.9% after the first 18ms. A reduction in Ω leads to the emergence of noise effects (the red curve), more than 98% of which are attributable to leakage into the {|u , |d } states.
Further speed-up of the gate is possible by increase in the radio wave Rabi frequency, while maintaining the constraint for noise suppression (18). This requirement is far from exhausted with the present simulation parameters. Simulation results for the single-qubit operations. The squared fidelity F 2 is plotted for each process, with the (unnormalised) input and comparison states shown. Results are obtained after averaging over 5 runs in each data set. TOP: Basic σy gate using Ω = 2π · 40 kHz (black), 2π · 5 kHz (red). Other parameters: φ−= φ+=1.57 rad, Ωg=2π ·177 Hz. MIDDLE: adiabatic transfer using Ω = 2π · 40 kHz (black), 2π · 6 kHz (red). Adiabatic rate is set to 1.57 rad/ms. BOTTOM: adiabatic σz gate using Ω = 2π · 24 kHz (black), 2π·103 kHz (red). Adiabatic rate is set at 1.57 rad/ms, so each data point is reached in 2ms.
On the other hand, slowing the gate down to the timescale of hundreds of milliseconds yields lifetimes that are consistent with the findings by Timoney et al. [18].
The gate is very robust against increased noise in the radio frequency field strength Ω g , yielding fidelity above 99% at 18ms even with f rw = 0.1 (using Ω = 2π · 40 kHz). An increase in the microwave noise to f = 0.01 is tolerable on the same criterion. In contrast, raising microwave noise to f = 0.05 yields a reduction to F < 90% at 18ms, which is uncorrectable even by raising Ω to 2π · 300 kHz. Figure 5 middle shows the simulation results for adiabatic transfer using a superposition state. Again, Ω = 2π · 40 kHz is found to provide sufficient shielding on the timescale considered, yielding 99.9% fidelity after the first 18ms, using a simulation average of 5 runs. Reduction in Ω leads to increase in noise effects, more than 97% of which are found to take the form of leakage out of the qubit space.

Adiabatic transfer
Simulations suggest that reliable adiabatic following is maintained for the adiabatic rate < Ω/20, so that speeding up of the process would eventually require an increase in the dressing field strength.
Simulations also suggest that f = 0.01 would represent a tolerable increase in microwave Rabi frequency noise, yielding fidelity F > 99% after 18ms (Ω = 2π · 40 kHz). On the other hand, setting f = 0.05 reduces the transfer fidelity below 90% at 18ms. This disturbance can not be corrected by raising Ω to 2π · 318 kHz. Figure 5 bottom panel displays results for the adiabatic σ z gate. In contrast to the other quantum operations, we have plotted the angular parameter x (see Figure 3) on the horizontal axis, since reaching any x takes the same amount of time for this gate. This property is also visible in the absence of any noise evolution with increasing x.

Adiabatic σz gate
The black curve illustrates a high-fidelity gate of 2ms duration (each point). As suggested in Section III D, an optimum value for Ω is indeed found, below and above which the gate fidelity is reduced. In this instance, using Ω = 2π · 24 kHz yields the fidelity of 99.9%, averaging over 5 runs. Reducing Ω leads to increased noise effects in the form of leakage. In contrast, using increased Ω is found to reduce leakage effects but to introduce disturbances of non-leakage type (this case is plotted in Figure 5 red curve).
Reducing the gate duration via increased adiabatic rate, while keeping all other parameters constant, does appear to lead invariably to better fidelities for the σ z gate. A limit on the gate speed is set by the adiabacity requirement, which is found to be: adiabatic rate < Ω/20.
Retaining the gate duration of 2ms but increasing the microwave and radio wave Rabi frequency noise to f, f rw = 0.05 leads to optimised fidelity with Ω in the vicinity of 2π · 6 kHz. Fidelities slightly above F = 95% are achieved.
Setting f, f rw = 0.01 enables one to reach fidelity above F = 99%, with optimum field strength in the vicinity of Ω = 2π · 11 kHz.

G. The effect of magnetic gradient
The multi-qubit entangling gate presented in Section IV makes intrinsic use of static magnetic-field gradient being present along the trap axis. This is also likely to be the case, within the experimental context, for the single-qubit gates. However, introducing a magnetic-field gradient in the single-qubit analysis of the present section is not expected to add a significant effect.
One can estimate analytically the magnitude of this contribution. Assuming a single motional mode only, the two phonon terms that would be added to the single-qubit Hamiltonian H (7) are: (see (40) and the definitions (38)). No sideband coupling is employed for the single-qubit gates, and one can view the total Hamiltonian in the interaction picture with respect to νb † b. This leaves the terms in H unaffected. Evaluating the magnitude of κσ z (b † + b) after the interaction picture, one recovers the following term in the second order: This would amount to a tiny effect for realistic experimental parameters (65). The effect on this term of a further interaction picture with respect to the microwave energy gap of form Ω (|u u| − |d d|) can be neglected, provided that Ω ν. Numerical simulation of single-qubit gates with magnetic gradient present has also been carried out (using parameters (65)) to establish that the gradient amounts to a negligible effect.

IV. MULTI-QUBIT GATE
It is now shown how the dressed-state approach, combined with magnetic-gradient-induced coupling [1], enables the realisation of an entangling gate. We consider the additional effect of static magnetic-field gradient along the trap axis and show how a Hamiltonian of Jaynes-Cummings form [33] can be obtained. It is then used to obtain the fast Mølmer-Sørensen gate [21].
Magnetic noise effects are discussed explicitly, demonstrating how microwave shielding can be accomplished. As the second key factor affecting the gate fidelity, we consider explicitly the effects of spurious couplings and resonances arising from the system Hamiltonian (39, 40) as well as the presence of an unused motional mode. Strategies for minimising these unwanted interactions are discussed. Further detrimental effects such as noise in the Rabi frequencies, effects due to stray addressing of individual particles in the frequency space, or known approximations of the trappedion physical system [34] could also be tackled in further research.
We derive and simulate an entangling gate for the twoparticle case with the simplification of considering explicitly a single motional mode only. The issue of avoiding coupling to the other motional mode is discussed, as well as the scope for extending the discussion to the multi-particle case.
A. Set-up and definitions Figure 6 depicts the arrangement for the gate implementation, together with definitions of the microwave and radio frequency fields. Two detuned radio frequency fields are employed, which generate four couplings in the {|−1 , |0 , |+1 } triplet of states. The two microwave fields required will be shown to generate a shielding effect directly analogous to that in the single-qubit gates. The presence of the magnetic field gradient makes the energies of |−1 and |+1 position-dependent, so that λ 0 now represents the equilibrium value of λ(z) for each trapped particle. Communication between individual qubits will be accomplished by means of the shared motional mode of the ions in the trap. The following additional variables are introduced: q − sideband detuning of radio waves (see Figure 6) ν − frequency of the shared motional mode n − phonon number b † , b − phonon operators, later redefined as: defined explicitly in the appendix C η = κ/ν − the effective Lamb-Dicke parameter R − integer parameter characterising the fast Mølmer-Sørensen gate (see (55))

B. Single-particle Hamiltonian
In the interaction picture with respect to H 0 = ω 0 |0 0|+ λ 0 |1 1| − λ 0 |−1 −1| and after performing the rotating wave approximation, one obtains the following Hamiltonian for the interactions depicted in Figure 6: The microwave and radio wave part (39) is directly analogous to the one previously quoted (7). Line (40) contains the phonon energy and the term due to the presence of the magnetic gradient [1]. As the next step, one applies the Schrieffer-Wolff transformation [35] of form: Its effect is to introduce factors to all terms in (39) as well as to remove the κσ z (b † + b) contribution. The following additional term is obtained after the transformation: Moving to the interaction picture with respect to the phonon term νb † b, one recovers the following Hamiltonian:

C. Jaynes-Cummings form
The gate will be illustrated for the case of the D-qubit, noting that an analogous construction for the B-qubit is possible. One sets in (43): Expanding the coupling terms to first order in η and changing basis to {|u , |d , |D , |0 }, one obtains: Line (46) gives the sought-after Jaynes-Cummings type of coupling in the qubit space. The terms oscillating with frequency ±q will be used in building the entangling gate, while the effect of the faster-oscillating ±(q+2ν) terms (47) will be minimised.
Line (45) is the energy gap created by the microwaves, analogous to the single-qubit case. H res represents numerous residual terms that contain ν and q in their rotation frequencies. An expression for H res in the interaction picture with respect to (45) is provided in the appendix (D1). These terms would be expected to cancel by rotating wave arguments, however, they will be shown to contribute to two distinct spurious coupling effects.
Considering the effect of magnetic noise in the dressed basis, the following contribution is found: This can be compared to (15). Moving to the interaction picture with respect to (45) will generate shielding against magnetic noise, as has been presented before. This mechanism is maintained, as one extends the discussion to multiparticle Hamiltonians.

D. Two-particle Hamiltonian
We present and simulate the entangling gate for the twoparticle case, noting that a multi-particle entangling gate would also be viable. The case discussed is for the D-qubit, using the centre-of-mass mode. The breathing mode is not treated explicitly, but the effects of its presence will be discussed in Section IV F.
The single-particle Hamiltonian (45-48) needs to be rederived for the extended (H ion1 ⊗ H ion2 ) ⊗ H phonon Hilbert space, making the necessary modifications. The term κσ z (b † + b) in line (40) enters with the same sign for each of the two qubits, provided that the centre-of-mass mode is assumed. One performs the Schrieffer-Wolff transformation of form: to remove the κσ zi (b † + b) contributions and recover the following extra term: The other steps in the derivation (interaction picture, rotating wave approximation, basis change) are generalised straightforwardly to the two-qubit case to yield a generalisation of the Hamiltonian (45-48). Finally, one moves to the interaction picture with respect to the (generalised version of) microwave part (45) to obtain the two-qubit Hamiltonian of the final form. This step leaves the terms (46-47) unaffected. Using the definition: the Jaynes-Cummings terms ((46), in the extended Hilbert space) can be rewritten in the form: This expression is used to obtain the fast Mølmer-Sørensen gate.
The effect of the faster-oscillating terms of form (47) (in the extended space) will be minimised by parameter choice. One checks for any other unwanted interactions in the final Hamiltonian by expanding it to the second order in the Dyson series and looking for secular terms. The following additional contribution is found: which affects significantly the gate performance and needs to be minimised.

E. Fast entangling gate
Following the proposal of Mølmer and Sørensen [21], a two-qubit entangling gate can be obtained from the Hamiltonian H q (53). The functions F (t) and G(t) (defined in the Mølmer-Sørensen derivation) need to be set to zero, which imposes the constraint: for integer R. Setting in addition: leads to the desired unitary evolution, which generates entanglement between the qubits: Given a value for R, the conditions (55, 56) fix the time of the entanglement operation to: Furthermore, the value for q is also determined:

F. Minimising spurious couplings
Experimental parameters have to be chosen to minimise excitations of the other motional mode and the effect of the resonance term (54). The breathing mode frequency is given by ν = √ 3ν (which is also the next lowest frequency in the N-particle case [36]), and the introduction of the breathing mode phonon terms ν b † b and ±κ σ z (b † + b ) in the Hamiltonian (see (40)) would lead to extra prefactors of form e ±η (b † −b ) in (43).
Considering the effect of such terms on the qubit-space couplings (46, 47), the next lowest oscillation frequency after e ±iqt will be close to e ±i(ν−ν )t (assuming q ν). It will be found in terms of the following functional form: where we have used ν = √ 3 ν and η = 3 −3/4 η (see appendix C). This represents the effect to be minimised, which generates a term in the second order of the Dyson series. Comparing this coupling with the strength of the gate coupling (58) leads to the condition: This constraint also ensures that the terms (47) yield a negligible effect. Secondly, the magnitude of the terms in (54) can be minimised (using the assumption ν 2 Ω 2 ) by requiring the following: Conditions (61) and (62), together with the expressions for T and q (58, 59) and the relationship η ∝ ν −3/2 constrain the choice of experimental parameters and ultimately the properties of the entangling gate that can be produced within a given set of experimental constraints. A good range of suitable parameters can still be found within reach of the current experimental capabilities ((65) provides an example).
The presence of a further motional mode in the derivation would also modify the expressions for H SW 2 (51) and H res (48), which would mathematically alter the unwanted resonance effects to some degree. This modification, which in general would depend on the particle number, can be tackled further by analytical and numerical techniques.

G. Fidelity correction
A third prominent unwanted coupling effect is found in numerical simulation and can be traced to terms in H res , specifically, the part proportional to Ω g (see (D1)). The effect of these terms is to superimpose a fast-oscillating time dependence on some of the plots for state fidelity during the gate operation.
The analytical treatment of this effect mirrors closely the derivation by Mølmer and Sørensen [21] (Section III A. Direct coupling). Firstly, we assume Ω q + ν, so that the terms responsible for the disturbance can be approximated to the following expression (here quoted for the single-particle Hamiltonian): Secondly, taking the desired gate evolution to be U (t), one transforms the disturbance (rewritten for the two-qubit case) to the interaction picture: H cI (t) = U † (t)H c (t)U (t), and considers expanding H cI (t) in the Dyson series to evaluate the magnitude of the disturbance. Two simplifying approximations are made. Firstly, U (t) is taken to be slowly-varying in comparison to H c (t), so that it can be regarded as constant when performing the Dyson series integrals. Secondly H cI (t) is evaluated in the vicinity of the endpoint of the gate operation (t = T ), where U (t) takes a simple form (57) and is approximated to be time-independent.
Obtaining an expression for H cI (t) in such a manner to the second order in the Dyson series, one can calculate the fidelity of certain output states, given a particular input state. One also needs to account for the fact that an interaction picture has been adopted. Again, we use a different definition of fidelity to the paper by Mølmer and Sørensen: F (|Ψ c , ρ) = Ψ c |ρ|Ψ c (see (35)), so that state probabilities are given by F 2 .
Beginning in the state |DD and calculating the fidelity of |DD at the end of the gate operation, one recovers F 2 = 1 2 . This is consistent with the unitary gate evolution (57) and is verified in the numerical simulation (see Figure 7 top), where no oscillatory effect is observed. In contrast, starting in the state |DD and calculating the fidelity of 1 √ 2 (|DD + i|0 0 ), the following is obtained: An oscillatory correction is thus introduced to the fidelity of the entanglement operation. Numerical simulation suggests that (64) predicts very accurately the frequency and the amplitude of the oscillations observed (see Figure 7 bottom). A simliar calculation can be carried out for any other input and comparison states. This oscillatory effect can be minimised by reducing Ω g /ν, or by adjusting precisely the gate duration. Higher trap frequency ν will lead to a greater accuracy requirement for the length of the gate pulse.

H. Simulation
Numerical simulation of the two-qubit entangling gate is carried out to demonstrate its feasibility. We simulate a Hamiltonian of the form (39)(40), extended to the two-qubit case. A single motional mode is used, which is chosen to be FIG. 7. TOP: Squared fidelity and other density matrix elements for the two-qubit entangling gate. An input state of |DD is used, and the simulation parameters are specified in (65). The first curve (counting from above at t ≈ 1.6 ms) represents the squared fidelity of |DD , where no oscillatory component is found. The second curve is the squared fidelity of 1 √ 2 (|DD + i|0 0 ). The third is the imaginary part of ρ |DD ,|0 0 , the fourth is the squared fidelity of |0 0 , and the last curve is the real part of ρ |DD ,|0 0 . BOTTOM: A magnified segment of the squared fidelity plot of 1 √ 2 (|DD + i|0 0 ) from the figure above. The result of the calculation (64) is plotted superimposed.
the centre-of-mass mode. The effects of magnetic noise in the multi-qubit case have been shown to be directly analogous to the single-qubit arrangement (see (49)), where sufficiently strong microwave dressing field renders the disturbance negligible. No magnetic noise or any other random noise effects have been included in the present simulation.
This parameter choice yields the gate time T = 3.7 ms, and sideband detuning q = 2π · 270 Hz. The constant of proportionality linking ν and η (see appendix C) is obtained for the 171 Y b + ion and magnetic gradient of 24 T/m. This represents well the current experimental capability [37]. Figure 7 plots squared state fidelities and density matrix elements for the duration of the gate operation. An input state of |DD has been used. The figure gives clear evidence for the feasibility of the entangling gate. Also, the oscillatory correction to the fidelity of the target state 1 √ 2 (|DD + i|0 0 ) is found to be in very good agreement with the mathematical description (64).
Numerical simulation also suggests that an increase in the phonon number to n = 5 would not introduce a significant reduction to the gate fidelity. This is in good agreement with the known properties of the Mølmer-Sørensen gate.

V. BEYOND THE LINEAR REGIME
This section discusses extensions and generalisations of the dressed-state approach to the regime where non-linear Zeeman shift plays a prominent role. The case of 171 Y b + is discussed in particular. We delineate precisely the 'Linear' regime for this physical system, which is the region of validity for the derivations presented above. We also define and discuss a 'Non-linear' regime, exemplified by the recent work of Webster et al. [19]. The relative merits of these two parameter ranges are then considered, together with a possible strategy for attaining either experimentally by means of microwave dressing fields.
A. Hyperfine Zeeman shift in 171 Y b + The four-level system depicted in Figure 1 can be realised using the F = {0, 1} hyperfine ground state of 171 Y b + with non-zero external magnetic field. The |1 and |−1 states would correspond to the m f = ±1 levels of the F = 1 triplet, F = 1, m f = 0 level would yield the |0 state and |0 would be represented by the singlet F = 0 state. The study by Blatt et al. [38] presents a detailed energy-level diagram of the system as well as provides an accurate measurement of the singlet-triplet energy splitting, which is approximately A = 2π · 12.6 GHz.
The |±1 states respond exactly linearly to external magnetic field B, with a change in energy of ±µ B B. The response of |0 and |0 can be approximated to the lowest order by ±(µ B B) 2 /A [39]. For any non-zero external field, there is therefore an inevitable discrepancy between the |−1 ↔ |0 and |0 ↔ |1 resonant frequencies, which can be well approximated by the (positive) figure: This enables the explicit definition of two simplified physical regimes.

B. Linear regime
The gates presented in the previous sections are built on the assumption of negligible ∆, so that addressing of both |−1 ↔ |0 and |0 ↔ |1 pairs can be achieved by the same Ω g field. Addressing one pair of levels exactly on resonance would mean that the other pair is addressed with the (positive) detuning equal to ∆. It is necessary to preserve this second coupling as a desired effect, with the contribution due to ∆ being negligible.
In the single-qubit case, considering the Rabi model [24], making the two interactions equivalent would require: (67) In the multi-qubit case, where the gate interaction strength is of the order ηΩ g , one requires ∆ to obey the following constraint: In both cases, an upper limit on the permissible magnetic field is placed by the strength of the RF fields employed. In the sections above, we have also assumed that magnetic noise affects prominently the {|−1 , |1 } states, but negligibly the {|0 , |0 } pair of levels. This relies on the assumption of small magnetic field. Comparing the sensitivity of |±1 to magnetic noise with the (B-field dependent) sensitivity of |0 leads to the requirement:

C. Non-linear regime
This regime is defined as the instance when both |−1 ↔ |0 and |0 ↔ |1 pairs can be unambiguously individually addressed, without affecting the other coupling. In this case, the coupling of the other pair, with the detuning equal to ∆, would represent an unwanted effect to be made negligible. This is the case for prominent ∆, such that the Stark shift approximation [29] applies. The condition is: which also ensures that the magnitude of the energy shift of |0 , Ω 2 g /4∆, is small compared to its Zeeman response, ∆/2, and therefore amounts to a negligible effect.
Experiments within the non-linear regime have been conducted by Webster et al. [19], also citing the condition (70). A field of 9.8 G is used to generate a measured frequency discrepancy ∆ = 2π · 29(1) kHz in agreement with (66). Radio frequency fields of strength Ω g = 2π · 1.9 kHz have been employed.
The authors have discussed how the non-linear regime enables the realisation of arbitrary single-qubit σ φ gates using a single radio frequency field. Also, the authors note that a σ z gate could be realised by the use of a single detuned radio field.
The facility of individual addressing does offer clear experimental advantages, however, it may also be the case that greater sensitivity to magnetic noise is introduced as well. Considering the criteria (67, 70), it is probable that the non-linear regime will involve stronger B-fields than the linear regime, particularly for the arrangement of an ion chain. If the condition (69) is broken, this would introduce non-negligible noise in the energy of |0 , which is not shielded against in the present set-up.
A further problem for the non-linear arrangement might arise in the attainment of individual addressing in an ion chain, due to the non-linear dependence of the energy spacings for individual qubits.

Single-qubit gates
A variety of ways to realise universal single-qubit rotations is possible in the non-linear regime. In addition to the proposals by Webster et al. [19], it is noted that individual addressing (φ − = φ + ) allows for the basic gate arrangement (Section III B) to yield both the σ x and the σ y gates for the B and D-qubits. An extra error source to consider would be the instability of the radio frequency fields (δ Ωg = Ω g− − Ω g+ ), due to two fields being necessary.
No extra effort would be required to realise adiabatic transfer, and the adiabatic σ z gate (Section III D) would be realisable by the usage of two RF fields per trapped particle. Further, the two σ z gates presented in the appendix are also a feasible alternative. In every case where two RF fields are being used, the small extra noise contribution due to δ Ωg would need to be considered.

Multi-qubit gate
The linear response of |−1 and |1 to magnetic field in the Ytterbium system permits the realisation of magneticgradient-induced coupling for any strength of the B-field, which is a crucial ingredient for the entangling gate. The reproduction of the Mølmer-Sørensen gate presented in this paper (Section IV) would be possible in the non-linear regime by the usage of four radio frequency fields per trapped particle.
Separate coupling of the magnetic-sensitive states is found to offer no clear mathematical advantage in the construction of the entangling gate. It is possible to employ two radio frequency fields (in two arrangements) and reach an entangling Hamiltonian of form similar to (45-48). However, the speed of the resultant gate is reduced by 1/2. Moreover, it is the property of the linear regime multiqubit gate that the zeroth order in η is cancelled within the qubit space, in the dressed basis, leaving only terms to the first order in η (see (46)(47)). This property ceases to hold for a gate that is built using two RF couplings per trapped particle. As a result, unwanted zeroth order terms of form Ω g e ±i(q+ν)t are introduced within the qubit space. This would lead to a more demanding set of constraints on the gate parameters.
These considerations make the Mølmer-Sørensen gate harder to realise in the non-linear regime.

D. Mediating technique
The linear and non-linear regimes are compounded by an intermediate region where neither perfect individual nor perfect mutual addressing in the qubit space are possible. The facility to reach either regime can be hampered by the existence of an upper limit on the B-field strength (69), as well as experimental limitations on the gate time or Ω g . In such cases, an intermediate regime may be inevitable, with the ensuing presence of spurious couplings within the qubit space.
As an alternative to tackling explicitly such couplings, the technique of dressed Stark shift (appendix A) offers a way of tuning ∆ by means of microwave fields. Such a process would potentially provide easy mediation between the linear and non-linear regimes. Using a detuned microwave field specified by Ω z , δ z to induce a |0 ↔ |0 coupling, together with the two microwave dressing fields, leads to the following additional term in ∆: subject to the conditions for fast oscillation (A4). This suggests the possibility of tuning ∆ with the help of a second physical process. The above result is found by considering Ω z and two microwave dressing fields only, so the potential cross-couplings due to the presence of RF fields would also need to be examined. Within an ion chain, it is likely that a single Ω z field would generate couplings between the |0 and |0 states of all the ions involved, so that no individual control over δ z and Ω z would be attainable. However, independent tuning of ∆ would still be possible, in principle, by means of the Ω dressing fields, which are well separated in frequency space.
Provided that the tuning of ∆ can be realised with attainable experimental parameters, dressed Stark shift offers a way of realising both linear and non-linear regimes using modest magnetic field strength. This would be of advantage for both single and multi-qubit designs.

VI. PROSPECTS FOR RADIO-WAVE-DRIVEN QUANTUM GATES
Section III B and the corresponding simulations (Section III F 1) have illustrated how a single-qubit gate of working time in the range of ms can be realised using RF fields of strength 2π · 177 Hz, and relying on the microwave dressing to provide the magnetic shielding effect.
One notes, in addition, that a scheme would also be possible, where the radio frequency fields both generate the gate coupling and provide magnetic shielding via the introduction of a time-dependent phase to the magnetic noise terms. Considering the D-qubit case (12) with only the Ω g component present, it is clear that rotations will be introduced to the magnetic noise term ((15), setting δ Ω = 0). A separate Dyson series analysis needs to be carried out to evaluate exactly the noise terms. However, one finds that, for such a set-up, noise suppression would occur for Ω g SD µ . It is found numerically that shielding is indeed accomplished, yielding robust gates on the timescale of µs.
In the case of the non-linear regime, this arrangement would permit the realisation of universal single-qubit rotation using radio-wave addressing only (see Section V C 1). It is an interesting research venue to pursue whether a feasible radio wave entangling gate could also be designed. In the absence of a viable shielding mechanism, it may be possible to out-pace the magnetic noise effects by realising a gate of sufficiently high speed.

VII. CONCLUSION
We have demonstrated the feasibility of universal quantum computing using microwave-dressed states in trapped ions or any other suitable system. Both single and multiqubit quantum operations have been proposed and their resilience against noise sources analysed in detail. This raises the prospects of microwave/radio wave-driven quantum computation as an exciting venue for future research. An interesting question to address would be the implementability of other entangling gate designs in the dressedstate system, possibly via adiabatic techniques.