Integrated photonic qubit quantum computing on a superconducting chip

We study a quantum computing system using microwave photons in transmission line resonators on a superconducting chip as qubits. We show that all control necessary for quantum computing can be implemented by coupling to Josephson devices on the same chip, and take advantage of their strong inherent nonlinearities to realize qubit interactions. We analyze the gate error rate to demonstrate that our scheme is realistic even for Josephson devices with limited decoherence times. A conceptually innovative solution based on existing technologies, our scheme provides an integrated and scalable approach to the next key milestone for photonic qubit quantum computing.

We study a quantum computing system using microwave photons in transmission line resonators on a superconducting chip as qubits. We show that all control necessary for quantum computing can be implemented by coupling to Josephson devices on the same chip. We take advantage of the strong nonlinearities inherent in Josephson junctions to realize qubit interactions. We analyze the gate error rate to demonstrate that our scheme is realistic even for Josephson devices with limited decoherence times. A conceptually innovative solution based on existing technologies, our scheme provides an integrated and scalable approach to the next key milestone for photonic qubit quantum computing. Despite the vast potential of quantum computers, no perfect physical implementation has been found for quantum computers. This can be seen by examining two representative systems. Josephson device based superconducting systems are easily integrable and scalable, but are plagued by the short decoherence times of Josephson qubits due to coupling to their complex solid-state environment. Photonic qubits, which have superb coherence properties, suffer from the fact that photons do not interact easily. Also, systems based on conventional bulk optical devices are hard to miniaturize and scale.
Recognizing the importance of integrated systems for scalable quantum computing, a number of investigators have demonstrated on-chip waveguide based quantum gates for photonic qubits recently [1]. This is a significant step that may represent the future direction of photonic qubit quantum computing technologies. However, it is a daunting task to achieve a fully integrated photonic qubit quantum computer using conventional technologies including those developed in the latest experiments. This is because conventional optical devices such as lasers, lenses, optical cavities and photo detectors are bulk devices based on very different technologies and no process exists yet to integrate them on the same chip [2]. Therefore, alternative realistic approaches to fully integrated photonic qubit quantum computing systems are highly valuable.
We combine the strengths of photonic and superconducting systems to realize fully integrated photonic qubit quantum computing. Our physical system is a superconducting chip on which high-Q transmission line resonators (TLRs) and Josephson devices are fabricated. The same system has been used for study of cavity QED based on Josephson qubits [3,4,5]. However, in our scheme the quantum information is carried by the microwave photon modes in the TLRs and the Josephson junctions play the role of optical devices. For high-Q TLRs the photons have a long life time [12] which is a major advantage. Easy operation and accurate control are available because Josephson devices can be fabricated with great precision and controlled conveniently by monitoring their electrical signals. A further key advantage is we can use the strong nonlinearities inherent in Josephson devices to induce interactions between photons. It is shown that high gate fidelities can be achieved even for Josephson devices with limited decoherence times making their unavoidable noisy environment no longer a limiting factor. Therefore, our scheme is a realistic approach to scalable photonic qubit quantum computing.
We start by considering the two identical TLRs shown in Fig.1(a). The TLR mode frequencies are given by ω = nπ/ √ LC, n an integer and L and C the total inductance and capacitance of the TLR. We use the n = 2 mode. For L = 0.5nH and C = 5pF, its frequency ω 0 /2π ≈ 20GHz. The second-quantized voltage and current associated with this mode is V (x, t) = ω 0 /C cos 2πx l (â(t)+â † (t)) and I(x, t) = −i ω 0 /L sin 2πx l (â(t)−â † (t)), where l the length of the TLR, x ∈ [−l/2, l/2] the position along the TLR, andâ(t) =âe −iω0t the mode's annihilation operator.  For the pair of identical TLRs in Fig.1(a), we introduce a single photon of frequency ω 0 and it being in the l eft or r ight TLR denotes the logic 0 or 1 state [6] for a single qubit. This is analogous to the conventional optical cavity mode representation of photonic qubit where the information is encoded by which cavity the photon is in [7]. Notice for a dilution refrigerator temperature of 40mK, the thermal photon number in the TLRs is smaller than 10 −10 and thus the 0 or 1 photon state for the TLRs is an excellent approximation. To effect arbitrary transformations on this single qubit, we need to be able to shift the relative energies of the TLRs and transfer photons between them, which implement the functionalities of phase shifters and beam splitters in optics. We realize this by coupling the TLRs capacitively to current biased Josephson junctions (CBJJ) as shown in Fig. 1 (a). As the simplest Josephson qubit, CBJJ has the advantage that its level splitting can be easily adjusted by the bias current. Approximating the CBJJs as two-state systems with adjustable energy splittings Ω c and Ω r [8], we can write , where σ z,± c,r are the Pauli matrices of the coupling and right CBJJ,â,b are the annihilation operators for photons in the two TLRs, and the coupling strengths the capacitance of the coupling and right CBJJ.
Since the CBJJ energies can be easily adjusted by tuning the bias current, we can control the interactions between the TLRs and CBJJs. To transfer photons between the TLRs, we adjust the bias currents of the CBJJs to tune Ω r faraway from ω 0 so the right CBJJ has no effect. We further tune the coupling CBJJ close to resonance with ω 0 and work in the dispersive region where the magnitude of detuning ∆ c = Ω c − ω 0 is much greater than g c . Assuming the CBJJ was prepared in the ground state, its virtual excitation gives rise to the following effective Hamiltonian for the TLRs [4] in the rotating frame defined by the uncoupled TLR Hamiltonian: Since there is only 1 photon in the system,â †â +b †b = 1, energy shifts described in the second term in H ef f is a constant. The first exchange term implements the photon transfer operation. A photon can be transferred between the two TLRs with a rate , which is about 20MHz for C c J = 0.5pF, C c = 23fF, and ∆ c /2π = 2GHz [10,11]. To shift the relative energies of the TLRs, we tune the coupling CBJJ far off resonance and tune the right CBJJ into the dispersive region. Similarly, an effective Hamiltonian ( g 2 r /∆ r )b †b results which gives a relative phase when the photon is in the right TLR.
We need to study the decoherence properties of our scheme to analyze its reliability. The photonic qubits have superb coherence and their life times are orders of magnitude longer than that of superconducting qubits. For TLRs fabricated on superconducting chips, a high quality factor of 10 6 − 10 7 has been demonstrated [12]. For TLR frequencies of tens of GHz, the photon loss rate κ/2π can be as low as KHz. In contrast, the CBJJ has a short decoherence time, and we assume its dephasing rate Γ 2 /2π ≈1MHz. The CBJJ's decay rate from the excited state Γ 1 /2π is on the order of 0.1MHz.
A major advantage of our scheme is that the relatively lossy CBJJ does not damp the coherence of the photonic qubits much since it is only virtually excited. The CBJJ's decay from the virtually excited state increases the photon's loss rate by (g c,r /∆ c,r ) 2 Γ 1 , which is not a concern since (g c,r /∆ c,r ) 2 Γ 1 is no greater than κ. To study the effect of the CBJJ's dephasing rate Γ 2 , we model the dephasing effect as the result of a random fluctuation δ n in the CBJJ's energy splitting. This introduces an uncertainty in the detuning during for instance a photon transfer operation, ∆ c = Ω c − ω 0 → ∆ c + δ n . Therefore, the system will have a random Hamiltonian H noise = − (g c /∆ c ) 2 δ n (â †b +âb † ) in addition to that in Eq. (1). Assuming the distribution of δ n is Gaussian, we can estimate the photonic qubit's decoherence time due to H noise by using the free induction decay function [13]. Since the free inductor decay function is determined by the spectral density of δ n , which in turn is related to the CBJJ's dephasing rate Γ 2 , it can be estimated that the system's dephasing rate is no greater than 2( Following the quantum theory of damping, we now calculate the gate error of a photon transfer operation under the influence of cavity loss and CBJJ dephasing. using the Master equation for the qubit's density matrix ρ, . The gate error probability of a single qubit bit flip is plotted in Fig. 1(b) as a function of κ and Γ 2 . The result indicates that, for already demonstrated Γ 2 /2π = 1MHz and κ/2π = 10kHz [14], the gate error is on the order of 10 −3 .
The manipulations demonstrated so far perform linear optics. We still need a mechanism to induce interactions between photons. This is a major difficulty in conventional optics. However, at microwave frequencies, we can take advantage of the strong nonlinearities in Josephson devices to interact photons.
We consider the low current biased 4-junction SQUID (FJS) device [15] in Fig. 2(a). The two small identical SQUIDs are coupled inductively to TLR C of length l at positions ±l/4. (This does not mean that the FJS must extend to a length of l/2 because the TLR can be layed out in a zig-zag fashion.) Since l is much larger than the dimension of the FJS, we can adopt the long wave approximation and use the TLR current at the SQUIDs' locations in calculating the SQUIDs' flux bias. At the In these equations, E c = (2e) 2 /4(C J + C s ) and E J = I c /2e are the charging energy and Josephson energy of the junctions, where C J and I c are the junctions' capacitance and critical current. φ is the average phase of the 4 junctions determined by the low bias current , L s and L L are the self-inductances of the small SQUIDs and the circuit loop. To simplify the expressions, we set χ = χ c = χ d and denote the photon frequency shift ω s ≡ ω c s = ω d s . The photon interaction strength ω int = −4E 0 J χ 4 cos φ/ . In deriving the system Hamiltonian, we have used the rotating wave approximation and dropped terms that will be oscillating fast in the rotating frame defined by H T LR . We have also dropped terms involving creation and annihilation of two photons. These terms have no effect since there is no more than 1 photon in the TLRs in our scheme.
We operate with a low bias current I b ≈ 0 for the FJS so that cosφ is large and the FJS' energy splitting is far away from the frequencies of the TLR photons. Thus, the FJS will not be excited by the TLR photons and they hardly get entangled. The FJS then acts as a "nonlinear medium" and Eq. (2) describes the interaction between photons in C and D modulated by the FJS' phase. For I c = 50µA, L s ≈ 10pH, and M C ≈ 80pH [16], the photon interaction strength ω int ≈ 1MHz, much greater than the photon loss rate. Unfortunately, there are difficulties in using this interaction for quantum computing. First, φ has fluctuations in it due to the FJS' charging energy and thus the interaction strength is not a constant. Also, it is not easy to turn off the interaction. Tuning φ close to π/2 requires biasing the FJS close to its critical current which makes the system unstable. The uncertainty in φ grows too.
To have the photons interact only when needed, we use a setup shown in Fig. 2 (c). Here TLRs A, B and E, F are two qubits with photons being in A and F representing their logic 0 state. When both qubits are in the 1 state, we can use the photon transfer operation discussed earlier to transfer the photons from B and E to the auxiliary TLRs C and D whose frequencies are made different than that of the qubit TLRs by ω s to account for their energy shifts. Once the photons are in C and D they can interact due to coupling to the FJS. Afterwards, we transfer them back to B and E.
To stabilize the FJS' phase, we shunt its junctions with large capacitances C s as shown in Fig. 2(a). At low bias currents the FJS' behavior can be very well approximated by that of a harmonic oscillator and the distribution of φ is given by its ground state If we choose a total capacitance C J + C s = 20pF, the relative uncertainty δ(ω int )/ω int ≈ 10 −4 . Such a small error is not a concern for the photon interaction term. However uncertainties in the photon energy shift terms ω s can be comparable to ω int and can cause large errors.
We employ a two-phase technique in the spirit of spinecho to address this problem. In phase 1, we first do a photon transfer operation between B, C and E, D with a speed relatively fast compared to ω int and δω s = −2E 0 J χ 2 δ(φ 2 )/ , the uncertainty in the photon frequency shift. We then wait for a desired interaction time t = π/ω int after which we do another photon transfer between B, C and E, D. In phase 2, we first perform a bit flip for the 2 qubits, in other words do a photon transfer operation between A, B and E, F . We then repeat phase 1. At the end, we perform a bit flip on the two qubits again. In this process, depending on their initial states the qubits will acquire the same random phase due to δω s in either phase 1 or 2, thus removing the effect of the randomness in the photon energy shifts. The end result is a π phase shift on the 2-qubit states if they are both in 0 or 1 initially. This is equivalent to a controlled phase gate and it enables universal quantum computing in combination with the single qubit operations.
If the photon transfer operation between B, C and E, D was perfect, the controlled phase gate would be exact. phase shift in C and D could be eliminated completely. However, since the photons in C and D will interact with the FJS even during the photon transfer, our control phase gate will have errors. This can be seen by examining the system Hamiltonian during the photon transfer (in the rotating frame) The first two terms are used for the photon transfer operation, however the remaining terms cannot be turned off making the photon transfer imperfect. Obviously, the fidelity of our controlled phase gate will be improved by making the photon transfer frequency λ bc and λ de large compared to δω s and ω int . We numerically studied our control phase gate using the full Hamiltonian and plotted the gate error in Fig 2 (b). We set λ ≡ λ bc = λ de . For our choice of system parameters, λ/δω s ≈ 20 and the gate error is on the order of 10 −3 .
Our microelectronic system is easily scalable as shown in Fig. 2 (c). We can extend the setup for the control phase gate in both ends to integrate many TLR qubits on the same chip with an FJS between each pair of qubits. This is a 1d architecture with controllable interactions between adjacent qubits that can be scaled to a large number of qubits.
In order to perform photonic qubit quantum computing, we still need to be able to generate and detect single photons. Photon generation on superconducting chip has been demonstrated experimentally [17,18,19]. For photon detection [20,21], we again consider a CBJJ coupled to a TLR of frequency ω 0 as shown in Fig. 3 (a). The CBJJ is prepared in the ground state |g in the well of its washboard potential. We also make use of an unstable excited state |e where the CBJJ can tunnel to the voltage state with a large rate Γ. By adjusting the CBJJ's bias current, we can tune the CBJJ in resonance with ω 0 . The CBJJ will then be excited by the TLR photon to |e . When it escapes from |e , a detectable voltage appears across the CBJJ.
Though an easy and reliable method, our scheme may fail to detect a photon in the TLR due to the photon decay and the CBJJ's intra-well decay and decoherence. The TLR photon may decay before being detected by the CBJJ. The CBJJ's intra-well decay from |e to |g and its finite decoherence time are concerns too. To study the influence of the photon loss rate and CBJJ's intra-well decay and decoherence on the ef- The dependence of detector efficiency(1−10 −P ) on the ratio between the escape rate Γ and photon loss rate κ. The coupling strength g td /2π = 100MHz, photon loss rate κ/2π = 10kHz, CBJJ decay and dephasing rate γT /2π = 100kHz, γϕ/2π = 1MHz. ficiency of our photon detector, we model it as a 3state system shown in Fig. 3 (a), where |f represents the voltage state. We use the Master equation dρ/dt = −i[H, ρ] + Lρ. Here, ρ is the density matrix of the system, the system Hamiltonian H = δâ †â + g td (â † σ ge +âσ eg ), the detuning δ = ω 0 − µ, µ the frequency difference between |e and |g . The Liouvillian κ is the decay rate of the photon in the TLR, γ T and γ ϕ are the intra-well decay rate and dephasing rate of the CBJJ, σ ij = |i j| for i, j = g, e, f , and σ z = |e e| − |g g|.
Assuming initially there is a photon in the TLR and the CBJJ is in |g , we plot the detecting efficiency (the probability the CBJJ ends up in |f ) in Fig. 3 (b) as a function of Γ/κ. As can be seen, the efficiency is high even for moderately large escaping rate Γ. For Γ/2π = 20MHz [22] and κ/2π = 10kHz, the detection efficiency is above 99%. Also, it is demonstrated in our simulation that the influence of γ ϕ to the detection efficiency is minor and thus the detecting CBJJ does not need to have long decoherence times.
In summary, we have shown that, by using TLR microwave photons as qubits and Josephson devices as optical devices, a superconducting chip provides an ideal implementation for fully integrated photonic qubit quantum computing. Thanks to our careful design, high gate fidelities can be achieved and thus our scheme is a realistic approach. Since our system is based on existing mature technologies, fast experimental progress can be expected to bring integrated photonic qubit quantum computing to reality. The novel idea of using on-chip microwave photons as qubits also opens the possibility of investigating many interesting optical quantum effects in an integrated system. This work was supported by NNSF of China (Grant Nos. 10875110, 60621064, 10874170) and National Fundamental Research Program of China (No.