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arXiv:1804.01127 · Physical Review A

Direct measurement of Bacon-Shor code stabilizers

Muyuan Li, Daniel Miller, Kenneth R. Brown

Published Nov 6, 2018• 27 citations
arXivHTMLPDF

Abstract

A Bacon-Shor code is a subsystem quantum error-correcting code on an $L\ifmmode\times\else\texttimes\fi{}L$ lattice where the $2(L\ensuremath{-}1)$ weight-$2L$ stabilizers are usually inferred from the measurements of $2L(L\ensuremath{-}1)$ weight-2 gauge operators. Here, we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance-3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped-ion quantum computer. We find that for a simple circuit of prepare, error correct, and measure, the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.

DOI
10.1103/physreva.98.050301

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A Bacon-Shor code is a subsystem quantum error-correcting code on an L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL lattice where the 2(L\ensuremath−1)2(L\ensuremath{-}1)2(L\ensuremath−1) weight-2L2L2L stabilizers are usually inferred from the measurements of 2L(L\ensuremath−1)2L(L\ensuremath{-}1)2L(L\ensuremath−1) weight-2 gauge operators. Here, we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance-3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped-ion quantum computer. We find that for a simple circuit of prepare, error correct, and measure, the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.
Quantum error correction is necessary to perform large-scale quantum computations in the presence of noise and decoherence. As a result, several aspects of quantum error correction have already been explored. These have been primarily studies of quantum memory[1, 2], an important first step towards quantum computation, where the objective is to increase the lifetime of the encoded quantum information. Additionally, several works have explored the implementation of logical gates[3-5]. In this work we study a next step - fault-tolerantly implementing quantum circuits. We choose the [[4,1,2]][[4, 1, 2]][[4,1,2]] Bacon-Shor subsystem code, which has a particularly simple error-detection circuit. Through both numerics and site-counting arguments, we compute pseudo-thresholds for the Pauli error rate ppp in a depolarizing noise model, below which the encoded circuits outperform the unencoded circuits. These pseudo-threshold values are shown to be as high as p=3%p=3\%p=3% for short circuits, and p=0.6%p=0.6\%p=0.6% for circuits of moderate depth. Additionally, we see that multiple rounds of stabilizer measurements give an improvement over performing a single round at the end. This provides a concrete suggestion for a small-scale fault-tolerant demonstration of a quantum algorithm that could be accessible with existing hardware.
The Bacon-Shor code is a quantum error correcting subsystem code composed of weight 2 check operators that admits a single logical qubit, and has distance ddd on a d×dd \times dd×d square lattice. We show that when viewed as a Floquet code, by choosing an appropriate measurement schedule of the check operators, it can additionally host several dynamical logical qubits. Specifically, we identify a period 4 measurement schedule of the check operators that preserves logical information between the instantaneous stabilizer groups. Such a schedule measures not only the usual stabilizers of the Bacon-Shor code, but also additional stabilizers that protect the dynamical logical qubits against errors. We show that the code distance of these Floquet-Bacon-Shor codes scales as Θ(d/k)Θ(d/\sqrt{k})Θ(d/k​) on an n=d×dn = d \times dn=d×d lattice with kkk dynamical logical qubits, along with the logical qubit of the parent subsystem code. Unlike the usual Bacon-Shor code, the Floquet-Bacon-Shor code family introduced here can therefore saturate the subsystem bound kd=O(n)kd = O(n)kd=O(n). Moreover, several errors are shown to be self-corrected purely by the measurement schedule itself. This work provides insights into the design space for dynamical codes and expands the known approaches for constructing Floquet codes.
We compare Steane's and Shor's syndrome extraction methods on the Bacon-Shor code. We propose a straightforward strategy based on postselection to prepare the logical ${|0\ensuremath{\rangle}}_{L}$ and ${|+\ensuremath{\rangle}}_{L}$ states of the Bacon-Shor code by using flaglike qubits to verify their constituent Greenberger-Horne-Zeilinger states. We perform stabilizer simulations with a circuit-level, depolarizing Pauli error model and find that Steane's method significantly outperforms Shor's. Not only does Steane's method result in pseudothresholds that are about one order of magnitude higher than Shor's, but also its advantage increases monotonically as we go from a distance-3 to a distance-9 Bacon-Shor code. The advantage of Steane's method is the greatest in the regime where gate errors dominate over measurement errors. Some of the circuit constructions we propose for Steane's method are not formally fault tolerant yet outperform the formally fault-tolerant Shor's protocols for experimentally relevant physical error rates. This suggests that constructing formally fault-tolerant circuits that maintain the full code distance is not strictly necessary to guarantee the usefulness of a quantum error-correcting protocol. Despite relying on postselection, we find that our methods can be efficient. These protocols would be naturally implementable on a platform with long-range qubit interactions like trapped ions or neutral atoms.
Quantum states decohere through interaction with the environment. Quantum error correction can preserve coherence through active feedback wherein quantum information is encoded into a logical state with a high degree of symmetry. Perturbations are detected by measuring the symmetries of the state and corrected by applying gates based on these measurements. To measure the symmetries without perturbing the data, ancillary quantum states are required. Shor error correction uses a separate quantum state for the measurement of each symmetry. Steane error correction maps the perturbations onto a logical ancilla qubit, which is then measured to check several symmetries simultaneously. We experimentally compare Shor and Steane correction of bit flip errors using the Bacon-Shor code implemented in a chain of 23 trapped atomic ions. We find that the Steane method produces fewer errors after a single round of error correction and less disturbance to the data qubits without error correction.
We construct two simple error correction schemes adapted to amplitude damping noise for Bacon-Shor codes and investigate their prospects for fault-tolerant implementation. Both consist solely of Clifford gates and require far fewer qubits, relative to the standard method, to achieve exact correction to a desired order in the damping rate. The first, employing one-bit teleportation and single-qubit measurements, needs only one-fourth as many physical qubits, while the second, using just stabilizer measurements and Pauli corrections, needs only half. The improvements stem from the fact that damping events need only be detected, not corrected, and that effective phase errors arising due to undamped qubits occur at a lower rate than damping errors. For error correction that is itself subject to damping noise, we show that existing fault-tolerance methods can be employed for the latter scheme, while the former can be made to avoid potential catastrophic errors and can easily cope with damping faults in ancilla qubits.
Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.