Logical qubits encoded in quantum error correcting codes can exhibit non-Markovian dynamical evolution, even when the underlying physical noise is Markovian. To understand this emergent non-Markovianity, we define a Markovianity condition appropriate to logical gate operations, and study it by relating logical operations to their physical implementation (operations on the data qubits into which the logical qubit is encoded). We apply our analysis to small quantum codes, and show that they exhibit non-Markovian dynamics even for very simple physical noise models. We show that non-Markovianity can emerge from Markovian physical operations if (and only if) the physical qubits are not necessarily returned to the code subspace after every round of QEC. In this situation, the syndrome qubits can act as a memory, mediating time correlations and enabling violation of the Markov condition. We quantify the emergent non-Markovianity in simple examples, and propose sufficient conditions for reliable use of gate-based characterization techniques like gate set tomography in early fault-tolerant quantum devices.
Comments
21 pages, 7 figures; removed unnecessary section from appendix A
Logical qubits encoded in quantum error correcting codes can exhibit non-Markovian dynamical evolution, even when the underlying physical noise is Markovian. To understand this emergent non-Markovianity, we define a Markovianity condition appropriate to logical gate operations, and study it by relating logical operations to their physical implementation (operations on the data qubits into which the logical qubit is encoded). We apply our analysis to small quantum codes, and show that they exhibit non-Markovian dynamics even for very simple physical noise models. We show that non-Markovianity can emerge from Markovian physical operations if (and only if) the physical qubits are not necessarily returned to the code subspace after every round of QEC. In this situation, the syndrome qubits can act as a memory, mediating time correlations and enabling violation of the Markov condition. We quantify the emergent non-Markovianity in simple examples, and propose sufficient conditions for reliable use of gate-based characterization techniques like gate set tomography in early fault-tolerant quantum devices.
As quantum devices are progressively scaled and refined, quantum codes will become indispensable to guarantee reliable computation in the presence of noise. Due to the difficulties of characterizing physical noise, direct measures of logical performance will be of paramount use to tune quantum codes [18]. Logical Randomized Benchmarking in [5] aimed to characterize the logical operations directly by performing randomized benchmarking using logical operators and following each operation with perfect QEC. However, due to resource limitations, physical implementations such as [14] only measure the syndrome at the end of a computation. \n \nWe develop a generalized framework for procedures that measure the fidelity of logical operations directly, logical fidelity estimation (LFE). We identify several freedoms that arise when attempting to define a fidelity which characterizes quantum error correction, specifically, in how error detection events are processed and in how often error correction is performed. We explore these freedoms by conducting a survey to eliminate inviable procedures guided by the presence of upticks, a standard signature of non-Markovianity [33]. Our survey highlights procedures that are robust to gate-dependent noise and procedures which are implementable in the near-term. We recommend applying post-selection after each gate as a benchmark for quantum codes which is robust to gate-dependent noise. We manage to eliminate some naive near-term protocols, but guaranteeing the robustness when QEC is applied at a at fixed time or when post-selection is applied at the end, are open problems.
Quantum error correction (QEC) can strongly suppress noise, but predicting whether a computation succeeds requires understanding not only how much effective noise persists after QEC, but what form that noise takes. To thoroughly characterize this noise, we introduce logical gate set tomography (``logical GST''). Logical GST produces detailed representations of noisy logical operations that can be used to derive higher-level metrics, such as gate fidelity and diamond distance, and to test for non-Markovian and coherent logical errors. We use logical GST to obtain the first full, self-consistent characterization of a universal single-qubit logical gate set, implemented in the seven-qubit Steane code using Quantinuum System Model H1, a 20-qubit trapped-ion quantum processor. In the experimental data from H1, logical GST identified predominantly logical Pauli-stochastic errors, found no evidence of logical non-Markovianity, and detected weak potential signatures of coherent logical error. By contrast, higher-shot datasets generated with Quantinuum's detailed system emulator, identify mechanisms for weak but statistically significant logical non-Markovianity and strong evidence of coherent logical error. Finally, we use logical GST to perform error attribution with the system emulator, revealing how specific physical error sources shape logical error mechanisms.
Fault-tolerant quantum computing requires extremely precise knowledge and control of qubit dynamics during the application of a gate. We develop a data-driven learning protocol for characterizing quantum gates that builds off previous work on learning the Nakajima-Mori-Zwanzig (NMZ) formulation of open system dynamics from time series data, which allows detailed reconstruction of quantum evolution, including non-Markovian dynamics. We demonstrate this learning technique on three different systems: a simulation of a qubit whose dynamics are purely Markovian, a simulation of a driven qubit coupled to stochastic noise produced by an Ornstein-Uhlenbeck process, and trapped-ion experimental data of a driven qubit whose noise environment is not characterized ahead of time. Our technique is able to learn the generators of time evolution, or the NMZ operators, in all three cases and can learn the timescale in which the qubit dynamics can no longer be accurately described by a purely Markovian model. Our technique complements existing quantum gate characterization methods such as gate set tomography by explicitly capturing non-Markovianity in the gate generator, thus allowing for more thorough diagnosis of noise sources.
Abstract The realization of fault-tolerant quantum computers hinges on effective quantum error correction (QEC) protocols, whose performance significantly relies on the nature of the underlying noise. In this work, we directly study the structure of non-Markovian correlated errors and their impact on surface code memory performance. Specifically, we compare surface code performance under non-Markovian noise and independent circuit-level noise, while keeping marginal error rates constant. Our analysis shows that while not all temporally correlated structures are detrimental, certain structures, particularly multi-time ‘streaky’ correlations affecting syndrome qubits and two-qubit gates, can severely degrade logical error rate scaling. Furthermore, we discuss our results in the context of recent QEC experiments on physical devices. These findings underscore the importance of understanding and mitigating non-Markovian noise toward achieving practical, fault-tolerant quantum computing.
Fault-tolerant quantum computing demands many qubits with long lifetimes to conduct accurate quantum gate operations. However, external noise limits the computing time of physical qubits. Quantum error correction codes may extend such limits, but imperfect gate operations introduce errors to the correction procedure as well. The additional gate operations required due to the physical layout of qubits exacerbate the situation. Here, we use density-matrix simulations to investigate the performance change of logical qubits according to quantum error correction codes and qubit layouts and the expected performance of logical qubits with gate operation time and gate error rates. Considering current qubit technology, the small quantum error correction codes are chosen. Assuming 0.1% gate error probability, a logical qubit encoded by a 5-qubit quantum error correction code is expected to have a fidelity 0.25 higher than its physical counterpart.
Abstract Fault-tolerant quantum computing requires many qubits with long lifetimes and accurate quantum gate operations. However, external noise limits the computing time and hampers accurate quantum gate operations. Quantum error correction (QEC) codes may extend such limits, but imperfect gate operations during QEC cause errors, which could cancel out QEC. We used density matrix simulations to examine the performance of QEC codes with five qubits. In current quantum devices, less than ten qubits are needed to conduct sufficient gate operations within their lifetime so that it is feasible to implement QEC codes. We analyzed the maximum tolerable error rate and error correction effect of individual QEC codes according to the qubit arrangement and gate accuracy. Assuming a 0.1% gate error probability, a logical ∣ 1 〉 state encoded by a five-qubit QEC code is expected to have a 0.25 higher fidelity than its physical counterpart.