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arXiv:2403.03291

Dynamical Logical Qubits in the Bacon-Shor Code

M. Sohaib Alam, Eleanor Gilbert Rieffel

Published Mar 5, 2024• 1 citation
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Abstract

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 $d$ on a $d \times d$ 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 $Θ(d/\sqrt{k})$ on an $n = d \times d$ lattice with $k$ 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 $kd = O(n)$. 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.

DOI
10.48550/arxiv.2403.03291

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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.
Quantum error correction (QEC) protects quantum systems against inevitable noises and control inaccuracies, providing a pathway toward fault-tolerant (FT) quantum computation. Stabilizer codes, including surface code and color code, have long been the focus of research and have seen significant experimental progress in recent years. Recently proposed time-dynamical QEC, including Floquet codes and generalized time-dynamical code implementations, opens up new opportunities for FT quantum computation. By employing a periodic schedule of low-weight parity checks, Floquet codes can generate additional dynamical logical qubits, offering enhanced error correction capabilities and, potentially, higher code performance. Here, we experimentally implement the Floquet-Bacon-Shor code on a superconducting quantum processor. We encode a dynamical logical qubit within a 3×3 lattice of data qubits alongside a conventional static logical qubit. We demonstrate FT encoding and measurement of the two-qubit logical states and stabilize these states using repeated error detection. We showcase universal single-qubit logical gates on the dynamical qubit. Furthermore, by implementing a logical controlled-not (cnot) gate, we entangle the dynamical and static logical qubits, generating an error-detected logical Bell state with a fidelity of 75.9%. Our results highlight the potential of Floquet codes for resource-efficient FT quantum computation.
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-2L2L2L 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.
We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pXp_XpX​ and the phase error probability pZp_ZpZ​, and determine how the probability of a logical error depends on pXp_XpX​ and pZp_ZpZ​. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased (pZ/pX≫1)p_Z/p_X\gg 1)pZ​/pX​≫1) and the physical error rate pZp_ZpZ​ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.
Floquet quantum error-correcting codes provide an operationally economical route to fault tolerance by dynamically generating stabilizer structures using only two-body Pauli measurements. But while it is well established that stabilizer codes in higher spatial dimensions gain additional levels of intrinsic robustness, higher-dimensional Floquet codes have hitherto been explored only in limited scope. Here we introduce a 3d generalization of a Floquet code whose instantaneous stabilizer group realizes a 3d fermionic toric code, while crucially preserving all three logical qubits throughout the entire measurement sequence. One central ingredient is the identification of a 3d lattice geometry that generalizes the features of the Kekulé lattice underlying the 2d Hastings-Haah code - specifically, a structure where deleting any one edge color yields a two-color subgraph that decomposes into short, closed loops rather than homologically nontrivial chains. This loop property avoids the collapse of logical information that plagues naive sequential two-color measurement schedules on many 3d lattices. Although, for our lattice geometry, a simple 3-round cycle that sequentially measures the three types of parity checks does not expose the full error syndrome set, we show that one can append a measurement sequence to extract the missing syndromes without disturbing the logical subspace. Beyond code design, 3d tricoordinated lattice geometries define a family of 3d monitored Kitaev models, in which random measurements of the non-commuting parity checks give rise to dynamically created entangled phases with nontrivial topology. In discussing the general structure of their underlying phase diagrams and, in particular, the existence of certain quantum critical points, we again make a connection to the general preservation of logical information in time-ordered Floquet protocols.
A central goal in quantum error correction is to reduce the overhead of fault-tolerant quantum computing by increasing noise thresholds and reducing the number of physical qubits required to sustain a logical qubit. We introduce a potential path towards this goal based on a family of dynamically generated quantum error correcting codes that we call "hyperbolic Floquet codes.'' These codes are defined by a specific sequence of non-commuting two-body measurements arranged periodically in time that stabilize a topological code on a hyperbolic manifold with negative curvature. We focus on a family of lattices for n qubits that, according to our prescription that defines the code, provably achieve a finite encoding rate ( 1 / 8 + 2 / n ) while still requiring only two-body measurements. Similar to hyperbolic surface codes, the distance of the code at each time-step scales at most logarithmically in n . The family of lattices we choose indicates that this scaling is achievable in practice. We develop and benchmark an efficient matching-based decoder that provides evidence of a threshold near 0.1% in a phenomenological noise model and 0.25% in an entangling measurements noise model. Utilizing weight-two check operators and a qubit connectivity of 3, one of our hyperbolic Floquet codes uses 400 physical qubits to encode 52 logical qubits with a code distance of 8, i.e., it is a [ [ 400 , 52 , 8 ] ] code. At small error rates, comparable logical error suppression to this code requires 5x as many physical qubits (1924) when using the honeycomb Floquet code with the same noise model and decoder.