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arXiv:2401.04530 · Physical Review A

Coherent errors in stabilizer codes caused by quasistatic phase damping

Dávid Pataki, Áron Márton, János K. Asbóth, András Pályi

Published Jul 8, 2024• 6 citations
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Abstract

Quantum error correction is a key challenge for the development of practical quantum computers, a direction in which significant experimental progress has been made in recent years. In solid-state qubits, one of the leading information loss mechanisms is dephasing, usually modeled by phase-flip errors. Here, we introduce quasistatic phase damping, a more subtle error model which describes the effect of Larmor frequency fluctuations due to $1/f$ noise. We show how this model is different from a simple phase-flip error model, in terms of multicycle error correction. Considering the surface code, we provide numerical evidence for an error threshold, in the presence of quasistatic phase damping and readout errors. We discuss the implications of our results for spin qubits and superconducting qubits.

DOI
10.1103/physreva.110.012417

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Quantum error correction is a key challenge for the development of practical quantum computers, a direction in which significant experimental progress has been made in recent years. In solid-state qubits, one of the leading information loss mechanisms is dephasing, usually modeled by phase-flip errors. Here, we introduce quasistatic phase damping, a more subtle error model which describes the effect of Larmor frequency fluctuations due to 1/f1/f1/f noise. We show how this model is different from a simple phase-flip error model, in terms of multicycle error correction. Considering the surface code, we provide numerical evidence for an error threshold, in the presence of quasistatic phase damping and readout errors. We discuss the implications of our results for spin qubits and superconducting qubits.
We consider the combined effect of readout errors and coherent errors, i.e., deterministic phase rotations, on the surface code. We use a recently developed numerical approach, via a mapping of the physical qubits to Majorana fermions. We show how to use this approach in the presence of readout errors, treated on the phenomenological level: perfect projective measurements with potentially incorrectly recorded outcomes, and multiple repeated measurement rounds. We find a threshold for this combination of errors, with an error rate close to the threshold of the corresponding incoherent error channel (random Pauli-Z and readout errors). The value of the threshold error rate, using the worst case fidelity as the measure of logical errors, is 2.6%. Below the threshold, scaling up the code leads to the rapid loss of coherence in the logical-level errors, but error rates that are greater than those of the corresponding incoherent error channel. We also vary the coherent and readout error rates independently, and find that the surface code is more sensitive to coherent errors than to readout errors. Our work extends the recent results on coherent errors with perfect readout to the experimentally more realistic situation where readout errors also occur.
Abstract Surface codes are building blocks of quantum computing platforms based on 2D arrays of qubits responsible for detecting and correcting errors. The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.
Abstract Realizing the potential of quantum computing requires sufficiently low logical error rates 1 . Many applications call for error rates as low as 10 −15 (refs. 2–9 ), but state-of-the-art quantum platforms typically have physical error rates near 10 −3 (refs. 10–14 ). Quantum error correction 15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device 18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.
Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.
We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes, we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the five-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}$ of the logical qubit with the same quantity at the physical level ${D}_{\ensuremath{\diamond}}$, we show that ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\le}c{D}_{\ensuremath{\diamond}}^{d}$ where ddd is the distance of the code and ccc is a constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance ddd will exhibit a scaling ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
This chapter discusses quantum noise and techniques for quantum error correction, a necessity for quantum computing. It discusses bit-flip errors, phase-flip errors, and their combination. The formalism of quantum operations is introduced, along with the operator-sum representation, and Krauss operators. With this, the chapter discusses the depolarization channel and imprecise gates, as well as (briefly) amplitude and phase damping. For error correction, repetition codes are introduced to motivate Shor’s 9-qubit error correction technique.