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arXiv:2012.07911 · Quantum

Relaxation times do not capture logical qubit dynamics

Amit Kumar Pal, Philipp Schindler, Alexander Erhard, ́Angel Rivas, Miguel-Angel Martin-Delgado, R. Blatt, Thomas Monz, Markus Müller

Published Jan 24, 2022• 0 citations
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Abstract

Quantum error correction procedures have the potential to enable faithful operation of large-scale quantum computers. They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings. To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed. In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fail and do not capture dynamics of logical qubits. We experimentally illustrate that, in particular, spatial noise correlations can give rise to rich and counter-intuitive dynamical behavior of logical qubits. We show that a suitable set of observables, formed by code space population and logical operators within the code space, allows one to track and characterize the dynamical behaviour of logical qubits. Awareness of these effects and the efficient characterisation tools used in this work will help to guide and benchmark experimental implementations of logical qubits.

DOI
10.22331/q-2022-01-24-632

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Quantum error correction procedures have the potential to enable faithful operation of large-scale quantum computers. They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings. To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed. In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fail and do not capture dynamics of logical qubits. We experimentally illustrate that, in particular, spatial noise correlations can give rise to rich and counter-intuitive dynamical behavior of logical qubits. We show that a suitable set of observables, formed by code space population and logical operators within the code space, allows one to track and characterize the dynamical behaviour of logical qubits. Awareness of these effects and the efficient characterisation tools used in this work will help to guide and benchmark experimental implementations of logical qubits.
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.
Quantum error correction procedures have the potential to enable faithful operation of large-scale quantum computers.They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings.To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed.In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fail and do not capture dynamics of logical qubits.We experimentally illustrate that, in particular, spatial noise correlations can give rise to rich and counter-intuitive dynamical behavior of logical qubits.We show that a suitable set of observables, formed by code space population and logical operators within the code space, allows one to track and characterize the dynamical behaviour of logical qubits.Awareness of these effects and the efficient characterisation tools used in this work will help to guide and benchmark experimental implementations of logical qubits.
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.
Decoherence-free subspaces protect quantum information from the effects of noise that is correlated across the physical qubits used to implement them. Given the ability to impose suitable Hamiltonians upon such a multi-qubit system, one can also implement a set of logical gates which enables universal computation on this information without compromising this protection. Real physical systems, however, seldom come with the correct Hamiltonians built-in, let alone the ability to turn them off and on at will. In the course of our development of quantum information processing devices based on liquid-state NMR, we have found the task of operating on quantum information encoded in decoherence-free subspaces rather more challenging than is commonly assumed. This contribution presents an overview of these challenges and the methods we have developed for overcoming them in practice. These methods promise to be broadly applicable to many of the physical systems proposed for the implementation of quantum information processing devices.
Rapidly developing experiments across multiple platforms now aim to realise small quantum codes, and so demonstrate a memory within which a logical qubit can be protected from noise. There is a need to benchmark the achievements in these diverse systems, and to compare the inherent power of the codes they rely upon. We describe a recently introduced performance measure called integrity , which relates to the probability that an ideal agent will successfully ‘guess’ the state of a logical qubit after a period of storage in the memory. Integrity is straightforward to evaluate experimentally without state tomography and it can be related to various established metrics such as the logical fidelity and the pseudo-threshold. We offer a set of experimental milestones that are steps towards demonstrating unconditionally superior encoded memories. Using intensive numerical simulations we compare memories based on the five-qubit code, the seven-qubit Steane code, and a nine-qubit code which is the smallest instance of a surface code; we assess both the simple and fault-tolerant implementations of each. While the ‘best’ code upon which to base a memory does vary according to the nature and severity of the noise, nevertheless certain trends emerge.
Quantum error correction protocols have been developed to offset the high sensitivity to noise inherent in quantum systems. However, much is still unknown about the behavior of a quantum error-correcting code under general noise, including noisy measurements. This lack of knowledge is largely due to the computational cost of simulating quantum systems large enough to perform nontrivial encodings. In this paper, we develop general methods for incorporating noisy measurement operations into simulations of quantum error-correcting codes and show that measurement errors on readout qubits manifest as a renormalization on the effective logical noise. We also derive general methods for reducing the computational complexity of calculating the exact effective logical noise by many orders of magnitude. This reduction is achieved by determining when different recovery operations produce equivalent logical noise. These methods could also be used to better approximate soft decoding schemes for concatenated codes or to reduce the size of a lookup table to speed up the error correction step in implementations of quantum error-correcting codes. We give examples of such reductions for the three-qubit, five-qubit, Steane, concatenated, and toric codes.