Abstract
The code capacity threshold for error correction using biased-noise qubits is known to be higher than with qubits without such structured noise. However, realistic circuit-level noise severely restricts these improvements. This is because gate operations, such as a controlled-NOT (CX) gate, which do not commute with the dominant error, unbias the noise channel. Here, we overcome the challenge of implementing a bias-preserving CX gate using biased-noise stabilized cat qubits in driven nonlinear oscillators. This continuous-variable gate relies on nontrivial phase space topology of the cat states. Furthermore, by following a scheme for concatenated error correction, we show that the availability of bias-preserving CX gates with moderately sized cats improves a rigorous lower bound on the fault-tolerant threshold by a factor of two and decreases the overhead in logical Clifford operations by a factor of five. Our results open a path toward high-threshold, low-overhead, fault-tolerant codes tailored to biased-noise cat qubits.
INTRODUCTION
With fault-tolerant quantum error correction (QEC), it is possible to perform arbitrarily long quantum computations provided that the error rate per physical gate or time step is below some constant threshold value and the correlations in the noise remain weak (1). Codes, such as the surface code, which involve only local operations, are attractive for practical realization. However, these codes come at the cost of demanding threshold requirements and prohibitively large overheads (2, 3). Current efforts in QEC are largely devoted to recovery from generic noise, which lacks any special structure. For example, in the widely studied depolarizing noise model, errors are represented with the stochastic action of the Pauli operators
Some efforts have been made toward designing QEC codes for biased-noise qubits (8–13). In particular, recent studies have shown ultrahigh code capacity thresholds for surface codes tailored to biased noise (12, 13). The code capacity is calculated by assuming noisy data qubits and noiseless syndrome-extraction circuits. However, errors during gate operations or circuit-level noise must be taken into account to estimate the fault-tolerant threshold. In the case of qubits with biased noise, operations that do not commute with the dominant error can unbias or depolarize the noise channel, reducing or eliminating any advantages conferred by the original biased noise.
To illustrate this point, consider first a system that preserves the noise bias. Suppose that we have a gate
Now, consider a controlled-NOT (CX) gate between the two qubits, implemented with an interaction of the form
Consequently, a phase-flip error is introduced in the control qubit depending on when the phase error on the target occurred. However, the phase flip of the target qubit during the gate propagates as a combination of phase flip and bit flip in the same qubit (for τ ≠ 0, T). Application of the CX gate therefore reduces the bias of the noise channel by introducing bit flips in the target qubit. In the same way, coherent errors in the gate operation arising from any uncertainty in V and T will also give rise to bit-flip errors in the target qubit. As a result, a native bias-preserving CX gate seems to be unphysical (8, 14). This is a serious drawback because the CX is a standard gate required to extract error syndromes in many error-correcting codes, including codes tailored to biased noise (12, 13). In the absence of a bias-preserving CX, alternate circuits are required for syndrome extraction. This was achieved in (8), for example, using teleportation schemes that require several CZ gates, measurements, and state preparations. The added complexity, however, limits the potential gains in fault-tolerant thresholds for error correction with biased-noise qubits.
Here, we show that a radical solution to the problem of implementing a bias-preserving CX exists with two-component cat qubits realized in a parametrically driven nonlinear oscillator (15). We choose to work in a basis in which the cat states
(A) Bloch sphere of the cat qubit. The figure also shows cartoons of the Wigner functions corresponding to the eigenstates of
Note that, in the limit of large α, the states
The cat states, or equivalently their superpositions, ∣0⟩ and ∣1⟩, are the degenerate eigenstates of a parametrically driven Kerr nonlinear oscillator (15). Compared to schemes based on harmonic oscillators (16–18), the advantage of the realization considered here is that the intrinsic Kerr nonlinearity, required to realize the cat qubit, also provides the ability to perform fast gates (19) [note that the Bloch sphere used in (19) is rotated from the one used here by 90o so that the Z and X axes are interchanged]. In addition, it has been theoretically shown that although phase flips increase linearly with the size of the cat α2, bit flips are exponentially suppressed (15, 20). As a result, this cat qubit exhibits a strongly biased noise channel. With these cats we show that it is possible to perform a native CX gate while preserving error bias. This gate is based on the topological phase that arises from the rotation of the cats in phase space generated by continuously changing the phase of the parametric drive. Because of the topological construction, the proposed CX gate preserves the error bias. Moreover, the noise channel of the gate also remains biased in the presence of coherent control errors. The ability to realize a bias-preserving CX gate differentiates the cat qubit from strictly two-level systems with biased noise and demonstrates the advantage of continuous-variable systems for fault-tolerant quantum computing.
This paper is organized as follows: We first describe the preparation of the driven cat qubit and present its error channel. We also discuss the implementation of trivially biased Z(θ) and ZZ(θ) gates. The ZZ(θ) gate can be used to reduce the overhead for magic-state distillation (10). We then show how the bias-preserving CX gate is implemented and provide the χ-matrix representation of the noisy gate. Lastly, to demonstrate the advantage of having physical bias-preserving CX gates, we analyze the scheme for concatenated error correction tailored to biased noise in (8). The scheme first uses a repetition code to correct for the dominant phase-flip errors. The overall noise strength after the first encoding is reduced compared to the unencoded qubits, and the effective noise strength is more symmetric. The repetition code is then concatenated with a Calderbank, Shor, and Steane (CSS) code. We find that the availability of a bias-preserving CX considerably simplifies the gadgets needed to implement fault-tolerant logical gates. Consequently, we are able to achieve an increase in the threshold by a factor of ≳2 and a reduction in the overhead by a factor of ≳5 for the repetition code gadgets.
RESULTS
Two-photon driven nonlinear oscillator
The Hamiltonian of a two-photon driven Kerr nonlinear oscillator in a frame rotating at the oscillator frequency ωr is given by
Here, K is the strength of the nonlinearity, while P and ϕ are the amplitude and phase of the drive, respectively, and
Since Eq. 5 commutes with the photon number parity operator, its eigenspace can be divided into even- and odd-parity subspaces, labeled in the figure by the red and blue levels, respectively. The degenerate cat subspace C (green) is separated from the rest of the Hilbert space
The phase ϕ of the two-photon drive is a continuous parameter that specifies the orientation of the cat in phase space. We define the cat qubit with the phase ϕ = 0 (see Fig. 1A), and for the discussion of the following two sections, we will fix this phase. As we will see in a few sections, this phase degree of freedom is, however, crucial for the implementation of the CX gate.
Dynamics in the qubit subspace
Suppose that a single-photon drive is applied to the oscillator at the resonance frequency ωr. In the rotating frame, the resulting Hamiltonian is
This expression shows that it is possible to implement an arbitrary rotation around the Z axis of the Bloch sphere using a single-photon drive by choosing θ = 0 (15, 20). The angle of rotation is determined by the strength J, duration of the single-photon drive, and the amplitude α of the cat state. Equation 7 also shows that, unlike rotation around the Z axis, rotation around the Y axis is suppressed exponentially with α2. That is, the external drive couples predominantly to
It is easy to extend the analysis above to realize a
The last line in the above equation is written in the limit of large α. In this limit, the term
Noise with narrow-band spectral density
Suppose that the oscillator couples to the environment with a general operator
From the analysis in the previous section, we see that
Thermal bath with narrow spectral density
By far, the main source of noise in oscillators is single-photon loss. In the cat subspace, one photon at a time is lost to the environment at frequency ωr. However, it is also possible for the oscillator to gain photons if the bath is at nonzero temperature. If the spectral density of thermal photons is narrow, but smooth and centered around ωr, then addition of a single photon to the oscillator (i.e., action of
For the first three error sources, single-photon dissipation, thermal noise with narrow spectral density, and pure dephasing with narrow spectral density, it is possible to obtain an analytical expression for the channel. The expressions for the coefficients in the limit when the product of rate of decoherence and time is small (i.e., κtα2 < 1 and κϕtα4 < 1) are given in the third column. Recall that
Narrow spectral density frequency noise
Apart from gain and loss of photons, it is possible that coupling with the environment causes the frequency of the oscillator to fluctuate. This noise channel is often referred to as pure dephasing. However, if these fluctuations are slow and of small amplitude compared to the energy gap, such as in the case of flux noise in superconducting circuits (25, 26), then the out-of-cat excitations are suppressed. Consequently, in the Born-Markov approximation, the Lindbladian is given by
As before, the last term is an approximation in the limit of large α. Table 1 shows the corresponding error channel in the limit of small κϕα4t.
Noise with wide-band spectral density
The previous section described the noise channel of the cat qubit coupled to a bath with narrow-band spectral density so that leakage is avoided. However, what if the spectrum of the environment-oscillator coupling is such that leakage out of the cat subspace becomes non-negligible? First, we will show that the leakage errors can be autonomously corrected by addition of photon dissipation. Second, we find that the amount of nondephasing errors introduced because of the autonomous correction process depends on the energy difference between the even- and odd-parity states of the mth excited manifold
Two-photon dissipation channel
In the presence of two-photon dissipation, the oscillator loses pairs of photons to the environment. The master equation of the parametrically driven oscillator in presence of a white two-photon dissipation channel is given by
The cat states
Thermal bath with white-noise spectrum
White thermal noise leads to the Lindbladian master equation,
(A) Addition of single photon at frequency ωr + Δωgap excites
Observe that the loss of two-photons causes transitions within the same parity subspace. Therefore, as illustrated in Fig. 2A, two-photon loss immediately after a single-photon gain event does result in phase flips. However, phase flips are already the dominant error channel in the system, and therefore, this effect does not change the structure of noise. However, the process of correcting leakage can also introduce bit flips. Before a two-photon jump event brings the population back to the cat manifold, the states
To confirm the analysis above, we numerically evaluate the error channel of the cat qubit as a function of α2 by simulating the master equation
The Hamiltonian
From the simulations, we find that the error channel takes the form
The coefficients λII, λIX, etc. are shown in Fig. 2B at time t = 50/K as a function of α for nth = 0.01, κ = K/400, and κ2ph = K/10. For a discussion on how the error channel is extracted from master equation simulations, see Methods. The time 50/K is chosen because it is the typical gate time on the stabilized cat qubit. As expected, for large ∣λIX∣, λXX, ∣λYZ∣, and λYY decrease exponentially with α2. The amount of leakage is quantified by
Similar to thermal noise, frequency fluctuations of the oscillator can also have a white spectral density. In the Supplementary Materials, we discuss the error channel for white frequency noise and provide numerical estimate for the corresponding error channel. As expected, we find that the nondephasing errors are suppressed exponentially with α2.
The analysis in this section can easily be extended to any form of incoherent and coherent (or control) errors. We can now summarize the results for a general environment-oscillator interaction. Suppose that the system operator that enters in the interaction Hamiltonian is of the form
Bias-preserving CX gate
As discussed earlier, for the noise channel to remain biased, the time-dependent unitary describing the system evolution during the gate must not explicitly contain an
Consider two cat qubits each stabilized in a two-photon driven Kerr nonlinear oscillator. The initial state of the system is
If the phase ϕ(t) is such that ϕ(0) = 0 and ϕ(T) = π, then at time T
As expected from the above discussion, a CX gate is realized by rotating the phase of the cat in the target oscillator by π conditioned on the control cat. The CX operation is based on the fact that during this rotation, the
Coupling with the environment during this evolution leads to errors in both the control and target cats. From the analysis earlier in section titled “Noise with wide-band spectral density”, the predominant stochastic errors are of the form
After this phase-flip event, the conditional phase continues to evolve and at time T
Therefore, a phase error on the control cat qubit at any time during the implementation of the CX is equivalent to a phase-flip on the control qubit after an ideal CX. Now, assume that a phase error occurred on the target at time τ. Immediately after this error, the state is
As before, after this phase-flip event, the conditional phase continues to evolve, and at time T
The above equations show that a phase-flip error on the target qubit at any time during the CX evolution is equivalent to phase errors on the control and target qubits after the ideal CX gate. That is, this CX gate based on rotation of the target cat qubit in phase space does not unbias the noise channel. This is in stark contrast with the CX gate implementation between two strictly two-level qubits and shows the advantage of using the larger Hilbert space of an oscillator. Although we have only explicitly showed the bias-preserving nature of the CX with respect to one phase flip in either the control or target cats, it is easy to extend the analysis above to multiple phase flips to see that the bias remains preserved. Moreover, note that any control errors in the target or control qubit can be expanded in the form
Suppose that the control error was such that at the end of the gate, ϕ(T) = π + Δ (instead of ϕ(T) = π). That is
Now,
Note that there is another source of rotation errors in the target cat. Any nondephasing error in the control qubit during the CX gate will cause leakage in the target oscillator. For example, a bit-flip error in the control cat at t = T/2 causes a phase-space rotation error in the target cat by π/2. That is, at the end of the gate, the target cat states are
Hamiltonian of the bias-preserving CX gate
Having seen that the evolution in Eq. 19 results in a CX gate with biased-noise error channel, we will now present the physical interaction Hamiltonian required to implement it. In general, we assume that the amplitudes of the cats in the target and control oscillators, α and β, respectively, are different. The following time-dependent interaction Hamiltonian implements the bias-preserving CX between the two oscillators
The first line in the above expression is the Hamiltonian of the parametrically driven nonlinear oscillator stabilizing the control cat qubit. The phase of the drive to this oscillator is fixed ϕ = 0. To understand the other two lines, recall that
Consequently, when the control qubit is in the state ∣0⟩, the state of the target oscillator remains unchanged. On the other hand, if the control qubit is in the state ∣1⟩ (∼ ∣−β⟩, for large β), then Eq. 26 is equivalent to
From the second term of this expression, we see that the cat states
The above equation shows that the last term of Eq. 28 leads to a dynamic phase, which exactly cancels the geometric phase. As a result, we find that when the control cat is in state ∣1⟩, an arbitrary state of the target qubit
Numerically simulated noise channel of the CX gate
To show that the Hamiltonian of Eq. 26 does result in a bias-preserving CX, we first simulate Eq. 26 without noise in the oscillators. We chose α = β = 2, ϕ(t) = πt/T, and T = 10/K. Figure 3 shows the Pauli transfer matrix obtained in this way. The infidelity between the CX resulting from the evolution under Eq. 26 and an ideal CX is as small as ∼9.3 × 10−7. This small infidelity, primarily resulting from nonadiabatic transitions due to finite KT, clearly shows that the Hamiltonian of Eq. 26 implements an ideal CX gate with an extremely high degree of accuracy.
The transfer matrix is obtained by simulating the Hamiltonian in Eq. 26 with α = β = 2, ϕ(t) = πt/T, and T = 10/K. The infidelity of this CX operation with respect to an ideal two-level CX is 9.3 × 10−7 and results from nonadiabatic transitions due to finite KT.
Next, to account for losses we numerically simulate evolution under the master equation
From this, we obtain the Pauli transfer matrix of the noisy CX,
For nth = 0, we find that the error channel is dominantly given by
For κ = K/4000, T = 10/K, and α = β = 2, λIcIt,IcIt ∼ 0.94, λZcIt,ZcIt ∼ 0.029, λIcZt,IcZt ∼ 0.015, λZcZt,ZcZt ∼ 0.015, λIcZt,ZcZt ∼ −0.009, and the gate fidelity is 94%. The leakage is 9.6 × 10−7, which does not notably increase from the case when losses are absent, and the bias is η ∼ 107.
Next, we obtain the error channel for nth = 1%. To correct for leakage two-photon dissipation
Lastly, we numerically estimate the error channel in case of overrotation. This can happen, for example, when control errors lead to the gate being implemented for slightly longer time T′ = T + δ(T). For the simulation, we choose πδ(T) = 0.01T corresponding to an overrotation of the target cat by an angle Δ = 0.01 (see Eq. 25). In this case, we simulate the master equation
For κ = K/4000, κ2 = K/5, and α = β = 2, λIcIt,IcIt ∼ 0.97, λZcIt,ZcIt ∼ 0.038, λIcZt,IcZt ∼ 0.015, λZcZt,ZcZt ∼ 0.024, λIcZt,ZcZt ∼ −0.009, and the bias is η ∼ 1955. For α = β = 2.2, the bias increases to η ∼ 2796. The above examples confirm that the noise channel of the CX gate is biased, and the bias increases with the size of the cat. Because of the large Hilbert space size, it becomes difficult to perform numerical simulations for larger α. However, using the insights from single oscillator simulations in the presence of thermal and frequency noise (see the Supplementary Materials), we expect to achieve a bias of ∼104 for α2 < 10 with experimentally reasonable experimental parameters.
Threshold and overhead for concatenation-based codes
To summarize the results so far, we have described the adiabatic preparation of the cat states
The idea introduced in (8) is to first encode the physical biased-noise qubits in a repetition code 𝒞1 and correct for dominant errors, in this case, phase flips. A repetition code with n qubits can correct (n − 1)/2 phase-flip errors. The code words are
Error correction in the repetition code
The (n − 1) stabilizer generators for the repetition code are
(A) Each blue shaded block is an error correction gadget for a repetition code with n = 3. The black and green lines indicate code and ancilla qubits, respectively. The green triangles facing the left and right represent preparation and measurement of the ancilla, respectively. In the naive scheme, (n − 1) stabilizer generators for the repetition code are measured using CX gates between pairs of data qubits and ancilla. Transversal CX gates between error-corrected code blocks (shown in the red shaded region) implement a
This decoding scheme is equivalent to constructing an r-bit repetition code for each of the (n − 1) stabilizer generators of the repetition code. Thus, each bit of syndrome from the inner code is itself encoded in an [r,1, r] repetition code so that decoding can proceed by first decoding the syndrome bits and then decoding the resulting syndrome. As we will see shortly, this naive way to decode the syndrome results in a simple analytic expressions for the logical error rates. However, it is by no means an ideal approach to decode, and one can imagine that the two-stage decoder above could be replaced by one that directly infers the most likely error on the n-qubit repetition code, given that s measured syndrome bits. In a few sections, we will explain the notion of a measurement code that exploits these insights to improve on the naive scheme by constructing a block code that can directly correct the bit-flip errors on the n data qubits in a single decoding step.
Logical CX gate (or CX ¯ ) with naive decoding
Since a physical CX with error channel biased toward dephasing errors is available, the
Each data qubit coming into the target and control blocks of the
Each of the error correction gadgets now measure (n − 1) syndromes, and each syndrome bit must be read correctly for successful decoding. Each syndrome bit is measured r times and requires two CX gates between a pair of code qubits and an ancilla. A syndrome measurement can be incorrect if the preparation or measurement of the ancilla was incorrect or if there was a dephasing error on the ancilla during the CXs. Therefore, an upper bound on the probability of error due to failure of the error correction in the target and control blocks is
In the worst case, a single nondephasing error occurring with probability ϵ/η anywhere in the circuit will cause the failure of the gadget. There are 4(n − 1)r CX gates in each of the error correction gadgets at the input and output and n transversal CX gates. As a result, the probability of an error due to a nondephasing fault is
Finally, the probability of a logical error in the
Figure 4 (B and C) compares the logical error rates for the
Moreover, we find that the
Recall that in the approach described above, the repetition code is concatenated with a CSS code. Therefore, εcat must be lower than the accuracy threshold for a CSS code for computation with arbitrarily high accuracy to be possible. For the example of the CSS code construction in (9, 28), the lower bound on the accuracy threshold is
Fault tolerance with a measurement code
As we discussed in the earlier section, the naive way to decode by measuring (n − 1) stabilizer generators is suboptimal. We will now discuss how we can improve decoding by using what we refer to as, a measurement code. To construct a measurement code, we desire that our syndrome measurement procedure measures a total of s elements of the stabilizer group (not necessarily the specified generators) by coupling to ancillas and that it can correct any t = (d − 1)/2 phase-flip errors on the n qubits. That is, we wish to have a classical code with parameters [n + s, n, d]. However, not every classical code with those parameters is admissible, because the classical parity checks must still be compatible with the stabilizers of the original quantum code, in this case, the repetition code. In particular, each parity check in the measurement code must have even weight when restricted to the data qubits so that it commutes with the logical
The general form of a measurement code can be specified by the parity check matrix HM. This, in turn, is specified as a function of the (generally redundant) parity checks HZ of the quantum repetition code and an additional set of s ancilla bits that label the measurements. Given HZ, the parity check matrix of the measurement code is the block matrix
The measurement of the jth parity check in the measurement code can be performed by a standard choice of circuit. We simply apply a CX gate to qubit i if there is a 1 in column i and target the ancilla labeled in column n + j. Note that by construction there is always a 1 in position (j, n + j) of HM. The effective error rate of this bare-ancilla measurement gadget will depend on the number of CX gates used and, hence, on the weight of the stabilizer being measured. Therefore, all other things (such as code distance) being equal, lower weight rows are preferred when designing a measurement code. Note that it is possible that the redundant stabilizers to be measured are higher weight or more nonlocal than the stabilizer generators themselves (see for example Eq. 41). In practice, because of experimental constraints, it may become more difficult to measure higher-weight/nonlocal stabilizers. However, this may be a vital tool to demonstrate fault-tolerant error correction and better than breakeven performance in near-term experiments.
The two examples we consider here are generated from the following choices for HZ, displayed here in transpose to save space
These codes were chosen to saturate the distance bound, so d = n for each code (so d = 3 and d = 5, respectively). These were found by guess work, and no attempt at finding optimal measurement codes was made, although these are the best of the few that were tested. To contrast our choices with the choice associated with repeating the measurements of the standard generators r times for n = r = 3, the measurement code is specified by
Both this choice and the n = 3 choice in Eq. 40 have distance d = 3 as measurement codes. However, our choice corresponds to a [6,3,3] measurement code, whereas the naive repeated generator method yields a [12,3,3] measurement code. In general, the naive scheme yields a [n + (n − 1)r, n, d(n, r)] code, and for smaller r, the distance will not yet saturate to n. For n = 5, we need r = 2 before the measurement code has distance 3 and r = 4 before the distance saturates at d = 5. Thus, the naive scheme yields either a [13,5,3] code or a [21,5,5] code, which are inferior in either distance or rate, respectively, to the [14,5,5] code that results from the choice in Eq. 41.
These examples also illustrate a counterintuitive feature of measurement codes. Consider again the naive repeated generator method with n = 5 and r = 2 or 4. If the decoder works by first decoding the syndrome bits individually, then the data are only protected against at most (r − 1)/2 = 0 or 1 arbitrary errors, respectively. However, a decoder that uses the structure of the associated measurement code can correct 1 or 2 arbitrary data errors with these respective parameters, which then reduces the leading order behavior of the code failure probability.
Both of the above codes in Eqs. 40 and 41 are small enough that the exact probability of a decoding failure can be computed via an exhaustive lookup table. To demonstrate the advantage of the measurement code over naive encoding and decoding, we estimate the probability of a logical error in the
DISCUSSION
Here, we have presented a driven cat qubit with highly biased noise channel and shown how to perform a CX gate, which preserves the error bias. A bias-preserving CX gate with strictly two-dimensional systems is impossible (8, 14). We are able to circumvent this no-go conjecture by exploiting the phase space topology of the underlying continuous variable system.
The physical realization of the CX gate requires a three-wave mixing between the oscillators. The natural coupling between two oscillators is, however, beam-splitter type. Fortunately, the oscillators are themselves fourth-order Kerr nonlinear. Thus, the required three-wave mixing can be generated by parametrically driving the target oscillator at a frequency ωd such that ωd = 2ωt − ωc. Here, ωt and ωc are the frequencies of the target and control oscillators, respectively. When this condition is satisfied, the fourth-order nonlinearity converts a photon in the drive and a photon in the control to two photons in the target. Thereby, an effective three-wave mixing is realized between the control and target. The Kerr nonlinearity of the oscillators themselves is sufficient to realize the CX interaction Hamiltonian, and no additional coupling elements are necessary. Moreover, because of the parametric nature, the coupling is controllable. A possible realization of the CX gate Hamiltonian in superconducting circuits is shown in Fig. 5. It is feasible to extend the scheme for the CX gate to implement a bias-preserving controlled-controlled-NOT (CCX) gate between three cat qubits. A naive circuit would, however, require a controllable four-wave mixing between the oscillators that is typically much weaker. As described in the Supplementary Materials, it is possible to implement a bias-preserving CCX gate using only three-wave mixing and four cat qubits. To summarize, the bias-preserving set of unitaries discussed in this paper, which are also physically implementable with three-wave mixing (or less), is {CX, CCX, ZZ(θ), Z(θ), CCZ}. These can be supplemented with state preparations P∣±⟩ and measurements
Here, the Kerr nonlinear oscillators (of frequencies ωt and ωc) are implemented with superconducting nonlinear asymmetric inductive elements or SNAILs (38, 39). A SNAIL can be biased with an external magnetic field so that it has both three- and four-wave mixing capabilities. It can therefore be used to implement the two-photon driven Kerr nonlinear oscillator and realize a cat qubit with biased-noise channel (19). The Hamiltonian in Eq. 26 can be simplified as
Furthermore, by adapting the scheme for concatenated error correction in (8), we have demonstrated that having bias-preserving CX gates leads to substantial improvements in fault-tolerant thresholds and overheads. At the level of repetition code, the estimated bound for fault-tolerant thresholds with naive decoding and experimentally reasonable biases of ∼103 − 104 is ∼0.55 % = 0.75%. Consequently, high-quality oscillators will still be required so that the phase-flip error remains small enough. One way to improve the threshold is by using better decoding techniques, for example, by using the measurement code. The approach based on concatenating a repetition code to another CSS code is not necessary or ideal. A more efficient technique would be to directly implement a code tailored to asymmetric noise such as the surface code (12, 13) or cyclic code (11) with the cat qubit. An analysis of these codes tailored to the cat qubits will be carried out in future work.
METHODS
Error channel from simulations
Here, we describe how the error channel in the sections titled “Thermal bath with white-noise spectrum” and “Numerically simulated noise channel of the CX gate” is extracted from master equation simulations. The dimension of the system of s cat qubits is d = 2s, and the elements of the Pauli transfer matrix R are
In the above expression, ℰ(·) is the error channel, and
CX gate in the presence of thermal noise
The error channel of the CX gate in the presence of thermal noise was given in the section titled “Numerically simulated noise channel of the CX gate.” The channel is obtained from the transfer matrix, which itself is obtained in two steps. First, the master equation for the CX,
As shown by Eq. 25, two-photon dissipation on the target oscillator
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/34/eaay5901/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).