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arXiv:2402.04093 · npj Quantum Information

Robust projective measurements through measuring code-inspired observables

Yingkai Ouyang

Published Oct 26, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Quantum measurements are ubiquitous in quantum information processing tasks, but errors can render their outputs unreliable. Here, we present a scheme that implements a robust projective measurement through measuring code-inspired observables. Namely, given a projective POVM, a classical code, and a constraint on the number of measurement outcomes each observable can have, we construct commuting observables whose measurement is equivalent to the projective measurement in the noiseless setting. Moreover, we can correct t errors on the classical outcomes of the observables’ measurement if the classical code corrects t errors. Since our scheme does not require the encoding of quantum data onto a quantum error correction code, it can help construct robust measurements for near-term quantum algorithms that do not use quantum error correction. Moreover, our scheme works for any projective POVM, and hence can allow robust syndrome extraction procedures in non-stabilizer quantum error correction codes.

DOI
10.1038/s41534-024-00904-y

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Abstract Quantum measurements are ubiquitous in quantum information processing tasks, but errors can render their outputs unreliable. Here, we present a scheme that implements a robust projective measurement through measuring code-inspired observables. Namely, given a projective POVM, a classical code, and a constraint on the number of measurement outcomes each observable can have, we construct commuting observables whose measurement is equivalent to the projective measurement in the noiseless setting. Moreover, we can correct t errors on the classical outcomes of the observables’ measurement if the classical code corrects t errors. Since our scheme does not require the encoding of quantum data onto a quantum error correction code, it can help construct robust measurements for near-term quantum algorithms that do not use quantum error correction. Moreover, our scheme works for any projective POVM, and hence can allow robust syndrome extraction procedures in non-stabilizer quantum error correction codes.
An important issue in the implementation of a quantum computer is to protect quantum information from decoherence. In fault-tolerant quantum computation, the circuits used to measure the error syndromes are themselves faulty; to minimize the effect of syndrome measurement errors, the syndromes are measured repeatedly. This paper introduces a scheme based on classical codes to make this process more robust and/or reduce the needed resources and measurement time. We analyze particular implementations based on low-density generator matrix (LDGM) codes using EXIT functions.
In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength. The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined. For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.
Determining the conditions under which positive operator-valued measures (POVMs), the most general class of quantum measurements, outperform projective measurements remains a challenging and largely unresolved problem. Of particular interest are projectively simulable POVMs, which can be realized through probabilistic mixtures of projective measurements, and therefore offer no advantage over projective schemes. Characterizing the boundary between simulable and non-simulable POVMs is, however, a difficult task, and existing tools either fail to scale efficiently, provide limited experimental feasibility or work only for specific POVMs. Here, we introduce and demonstrate a general method to certify non-simulability of a POVM by introducing a complete hierarchy of semidefinite programs. It provides upper bounds on the non-simulability measure of critical visibility of arbitrary POVMs which are tight in many cases and outperform previously known criteria. We experimentally certify the non-simulability of two- and three-dimensional POVMs using a trapped-ion qudit quantum processor by constructing non-simulability witnesses and introduce a modification of our framework that makes them robust against state preparation errors. Finally, we extend our results to the setting where an additional ancilla system is available.
We present an algorithm for manipulating quantum information via a sequence of projective measurements. We frame this manipulation in the language of stabilizer codes: a quantum computation approach in which errors are prevented and corrected in part by repeatedly measuring redundant degrees of freedom. We show how to construct a set of projective measurements which will map between two arbitrary stabilizer codes. We show that this process preserves all quantum information. It can be used to implement Clifford gates, braid extrinsic defects, or move between codes in which different operations are natural.
Quantum measurement is performed by coupling a quantity A ^ of a system to a “meter” and reading the meter. In the regime of strong measurement, the possible outcomes and the final state are entirely defined by A ^ , with probabilities dictated by the initial state. In this work, we will be interested in the regime in which a strong self-Hamiltonian, and not the coupling, defines the measurement outcomes and the final state. In this regime, the expectation value of any arbitrary set of observables can be measured in a quantum nondemolition (QND) way without reset or repreparation of the state. We discuss applications that include single-shot measurement of the expectation value of all noncommuting observables from a single copy of the state, without any statistics or ensembles, and the decoding of dephasing errors in a Schrödinger-cat qubit quantum error-correction (QEC) code. In the latter, the error syndrome is obtained from single-shot QND measurements of the mean photon number of the cat, even if neither the cat code words nor the stabilizing Hamiltonian commute with the photon-number operator.
Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. In this work, I revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. I describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigen states. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction is specific to each system-apparatus pair—a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit, requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. I analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter. I test the trajectory ensemble distribution predicted by the quantum diffusion model against the experimental data for weak measurement of superconducting transmon qubits. There is a good fit between theory and experiment for different initial states and several weak measurement couplings. This test vindicates the continuous stochastic measurement framework for quantum state collapse, where the rate of collapse is a characteristic parameter for each system-apparatus pair and is not a universal constant. Furthermore, it implies that the environment can influence the measurement outcomes only via the apparatus and not directly. These are important clues in construction of a complete theory of quantum measurement. The framework of weak measurements can also be used to construct quantum error correction protocols that protect a quantum state from external disturbances. Unlike projective measurements, one can extract only partial information about the error syndrome from the encoded state using weak measurements. I construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, I show that the error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshold beyond which multiple errors dominate, and (b) the…