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Arbitrary Nondemolition Measurements through Hamiltonian Stabilization for Quantum Error Correction

Rodrigo G. Cortiñas

Published Jun 20, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1103/prxquantum.6.020355

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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.
We derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.
A central requirement for any quantum error correction scheme is the ability to perform quantum nondemolition measurements of an error syndrome, corresponding to a special symmetry property of the encoding scheme. It is in particular important that such a measurement does not introduce extra error mechanisms, not included in the error model of the correction scheme. In this Letter, we ensure such a robustness by designing an interaction with a measurement device that preserves the degeneracy of the measured observable. More precisely, we propose a scheme to perform continuous and quantum nondemolition measurement of photon-number parity in a microwave cavity. This corresponds to the error syndrome in a class of error correcting codes called the cat codes, which have recently proven to be efficient and versatile for quantum information processing. In our design, we exploit the strongly nonlinear Hamiltonian of a high-impedance Josephson circuit, coupling a high-Q cavity storage cavity mode to a low-Q readout one. By driving the readout resonator at its resonance, the phase of the reflected or transmitted signal carries directly exploitable information on parity-type observables for encoded cat qubits of the high-Q mode.
Quantum error correction (QEC) codes are traditionally defined and searched for without specifying the manner in which its syndrome extraction circuits are executed using elementary gates and measurements. We show how morphing circuits introduced in Refs. [1-3] provide a way of optimising syndrome extraction circuits and codes directly in terms of connectivity, choice of two-qubit gate (ISWAP versus CNOT) and number of physical qubits. We discuss morphing circuits in code optimisation among Abelian two-block group algebra (2BGA) codes, handling boundaries for 2D codes, codes with single-shot properties, and improving performance in stability experiments against measurement and reset errors. We show that alternating syndrome extraction circuits - executed with alternating time-reversed rounds - can be viewed as a two-round morphing circuit whose fault-tolerant properties are computationally much easier to examine than non-alternating syndrome extraction circuits. Our methods find new codes and syndrome extraction circuits of practical interest, including Abelian 2BGA morphing circuits with better code parameters and connectivity than existing circuits. [1] Matt McEwen, Dave Bacon, and Craig Gidney. Relaxing hardware requirements for surface code circuits using time-dynamics. Quantum, 7:1172, 2023. [2] Craig Gidney and Cody Jones. New circuits and an open source decoder for the color code, 2023. [3] Mackenzie H. Shaw and Barbara M. Terhal. Lowering connectivity requirements for bivariate bicycle codes using morphing circuits.
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.
Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requiresΩ(rd2)\Omega (r d^{2})Ω(rd2)parity check measurements for a distance-dddcode defined byrrrparity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using onlyrrrmeasurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rateppp. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved usingO(dlog⁡(d))O(d \log (d))O(dlog(d))measurements for any code with distanced≥Ω(nα)d \geq \Omega (n^\alpha)d≥Ω(nα)for some constantα>0\alpha > 0α>0. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer thanrrrmeasurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.
In this paper we investigate stabilizer quantum error correction codes using controlled phase rotations of strong coherent probe states. We explicitly describe two methods to measure the Pauli operators that generate the stabilizer group of a quantum code. First, we show how to measure a Pauli operator acting on physical qubits using a single coherent state with large average photon number, displacement operations, and photon detection. Second, we show how to measure the stabilizer operators fault-tolerantly by the deterministic preparation of coherent quantum superposition (``cat'') states along with one-bit teleportations between a qubitlike encoding of coherent states and physical qubits.