Modern quantum devices are highly susceptible to errors, making the verification of their correct operation a critical problem. Usual tomographic methods rapidly become intractable as these devices are scaled up. In this paper, we introduce a general framework for the efficient verification of large quantum systems. Our framework combines robust fidelity witnesses with efficient classical post-processing to implement measurement back-propagation. We demonstrate its usefulness by focusing on the verification of bosonic quantum systems, and developing efficient verification protocols for large classes of target states using the two most common types of Gaussian measurements: homodyne and heterodyne detection. Our protocols are semi-device independent, designed to function with minimal assumptions about the quantum device being tested, and offer practical improvements over previous existing approaches. Overall, our work introduces efficient methods for verifying the correct preparation of complex quantum states, and may be used for calibrating large quantum devices, witnessing quantum properties, supporting demonstrations of quantum computational speedups and enhancing trust in quantum computations.
Modern quantum devices are highly susceptible to errors, making the verification of their correct operation a critical problem. Usual tomographic methods rapidly become intractable as these devices are scaled up. In this paper, we introduce a general framework for the efficient verification of large quantum systems. Our framework combines robust fidelity witnesses with efficient classical post-processing to implement measurement back-propagation. We demonstrate its usefulness by focusing on the verification of bosonic quantum systems, and developing efficient verification protocols for large classes of target states using the two most common types of Gaussian measurements: homodyne and heterodyne detection. Our protocols are semi-device independent, designed to function with minimal assumptions about the quantum device being tested, and offer practical improvements over previous existing approaches. Overall, our work introduces efficient methods for verifying the correct preparation of complex quantum states, and may be used for calibrating large quantum devices, witnessing quantum properties, supporting demonstrations of quantum computational speedups and enhancing trust in quantum computations.
Quantum error correction is pivotal in advancing toward large-scale quantum computation, and efficient verification is crucial for ensuring the high fidelity of code states. Traditional methodologies, such as state tomography, direct fidelity estimation, and state verification, either fall short in measurement efficiency, especially for large-scale systems, or are restricted to some specific states. In this work, we introduce a general framework for quantum subspace verification, enabling efficient and measurement-noise-robust fidelity estimation between a given state and the target subspace with a specified confidence level. By integrating the proposed subspace verification with direct fidelity estimation, we develop a composite protocol that significantly improves the efficiency of verifying the fidelity of general magic logical states, as demonstrated by intuitive numerical results. This improvement stems from the use of subspace verification, which leverages the knowledge of code subspaces to significantly reduce measurement costs. Additionally, we detail the construction of verification operators for typical error correction codes, including general stabilizer codes and quantum low-density parity-check codes, enabling their efficient implementation using practical local measurements. Notably, for certain codes, such as the Calderbank-Shor-Steane codes and quantum low-density parity-check stabilizer codes, we reduce the number of required measurement settings and sample complexity to a constant level using graphical methods. Our approach facilitates efficient and feasible verification of error correction codes and generic magic logical states, advancing their practical implementations on quantum platforms.
Abstract We aim to devise feasible, efficient verification schemes for bosonic channels. To this end, we construct an average-fidelity witness that yields a tight lower bound for average fidelity plus a general framework for verifying optimal quantum channels. For both multi-mode unitary Gaussian channels and single-mode amplification channels, we present experimentally feasible average-fidelity witnesses and reliable verification schemes, for which sample complexity scales polynomially with respect to all channel specification parameters. Our verification scheme provides an approach to benchmark the performance of bosonic channels on a set of Gaussian-distributed coherent states by employing only two-mode squeezed vacuum states and local homodyne detections. Our results demonstrate how to perform feasible tests of quantum components designed for continuous-variable quantum information processing.
Preparation and verification of specific quantum states is an important capability for quantum devices to realise advantages over classical computations and algorithms. In this work, we have demonstrated an end-to-end framework that combines resource-efficient quantum state preparation with rapid, robust fidelity verification on near-term quantum hardware. By experimentally preparing and validating a structured complex quantum state encoding a digitized acoustic signal on the Quantinuum H2-1 trapped-ion platform, we achieved a high hardware fidelity of Fhw=0.929. Crucially, this milestone was realized without relying on idealized assumptions or deep fault-tolerant overhead, but rather through resource-minimal circuits optimized for NISQ-era and early fault-tolerant devices.
Furthermore, we addressed a key limitation in current quantum state certification. While validation methods like shadow overlap work well for random states, their sample complexity can become prohibitively high for the structured states used in practical algorithms. We mitigate this by introducing a pre-measurement basis-change technique that reduces the verification parameter, τ, by over 10 orders of magnitude for structured targets. This approach tightens the theoretical certification guarantees of the shadow overlap method and integrates tensor-network preparation and shadow validation into a unified workflow. These results shift the paradigm of how structured classical data can be mapped to and verified on quantum hardware under realistic noise and measurement budgets. By compressing a robust verification procedure to just 1,000 measurement shots, this framework offers an immediate, scalable benchmarking standard.
Efficient and reliable verification of quantum states is central to quantum information processing applications. If using well-characterized measurement devices, effective methods have been developed for quantum state verification (QSV). In reality, however, measurement devices are generally imperfect or untrusted, which limits significantly the application of standard QSV protocols. Here, we propose the measurement-device-independent QSV (MDI-QSV) scheme for practice. With the help of trusted quantum inputs, we have developed a systematical approach to design MDI-QSV strategies for an arbitrary pure target state. We find that the number of required measurements has an optimal scaling with required accuracy and confidence level, similar to the standard QSV where trusted measurement devices are available. Our results offer a sample-efficient and realistic method for quantum state verification with virtues of a measurement-device-independent manner, and are within reach of current technology.
The demonstration of quantum speedup, also known as quantum computational supremacy, that is the ability of quantum computers to outperform dramatically their classical counterparts, is an important milestone in the field of quantum computing. While quantum speedup experiments are gradually escaping the regime of classical simulation, they still lack efficient verification protocols and rely on partial validation. Here we derive an efficient protocol for verifying with single-mode Gaussian measurements the output states of a large class of continuous-variable quantum circuits demonstrating quantum speedup, including Boson Sampling experiments, thus enabling a convincing demonstration of quantum speedup with photonic computing. Beyond the quantum speedup milestone, our results also enable the efficient and reliable certification of a large class of intractable continuous-variable multimode quantum states.
Quantum computers promise to offer a considerable speed-up in solving certain problems, compared to the best classical algorithms. In many instances, the gap between quantum and classical running times is conjectured to be exponential. While this is great news for those applications where quantum computers would provide such an advantage, it also raises a significant challenge: how can classical computers verify the correctness of quantum computations? In attempting to answer this question, a number of protocols have been developed in which a classical client (referred to as verifier) can interact with one or more quantum servers (referred to as provers) in order to certify the correctness of a quantum computation performed by the server(s). These protocols are of one of two types: either there are multiple non-communicating provers, sharing entanglement, and the verifier is completely classical; or, there is a single prover and the classical verifier has a device for preparing or measuring quantum states. The latter type of protocols are, arguably, more relevant to near term quantum computers, since having multiple quantum computers that share a large amount of entanglement is, from a technological standpoint, extremely challenging. Before the realisation of practical single-prover protocols, a number of challenges need to be addressed: how robust are these protocols to noise on the verifier's device? Can the protocols be made fault-tolerant without significantly increasing the requirements of the verifier? How do we know that the verifier's device is operating correctly? Could this device be eliminated completely, thus having a protocol with a fully classical verifier and a single quantum prover? Our work attempts to provide answers to these questions. First, we consider a single-prover verification protocol developed by Fitzsimons and Kashefi and show that this protocol is indeed robust with respect to deviations on the quantum state prepared by the verifier. We show that this is true even if those deviations are the result of a correlation with the prover's system. We then use this result to give a verification protocol which is device- independent. The protocol consists of a verifier with a measurement device and a single prover. Device-independence means that the verifier need not trust the measurement device (nor the prover) which can be assumed to be fully malicious (though not communicating with the prover). A key element in realising this protocol is a robust technique of Reichardt, Unger and Vazirani for testing, using non-local correlations, that two untrusted devices share a large number of entangled states. This technique is referred to as rigidity of non-local correlations. Our second result is to prove a rigidity result for a type of quantum correlations known as steering correlations. To do this, we first show that steering correlations can be used in order to certify maximally entangled states, in a setting in which each test is independent of the previous one. We also show that the fidelity with which we characterise the state, in this specific test, is optimal. We then improve the previous result by removing the independence assumption. This then leads to our desired rigidity result. We make use of it, in a similar fashion to the device-independent case, in order to give a verification protocol that is one-sided device-independent. The importance of this application is to show how different trust assumptions affect the efficiency of the protocol. Next, we describe a protocol for fault-tolerantly verifying quantum computations, with minimal "quantum requirements" for the verifier. Specifically, the verifier only requires a device for measuring single-qubit states. Both this device, and the prover's operations are assumed to be prone to errors. We show that under standard assumptions about the error model, it is possible to achieve verification of quantum computation using fault-tolerant principles. As a proof of principle, and to better illustrate the inner workings of the protocol, we describe a toy implementation of the protocol in a quantum simulator, and present the results we obtained, when running it for a small computation. Finally, we explore the possibility of having a verification protocol, with a classical verifier and a single prover, such that the prover is blind with respect to the verifier's computation. We give evidence that this is not possible. In fact, our result is only concerned with blind quantum computation with a classical client, and uses complexity theoretic results to argue why it is improbable for such a protocol to exist. We then use these complexity theoretic techniques to show that a client, with the ability to prepare and send quantum states to a quantum server, would not be able to delegate arbitrary NP problems to that server. In other words, even a client with quantum capabilities cannot exploit those capabilities to delegate the computation of NP problems, while keeping the input, to that computation, private. This is again true, provided certain complexity theoretic conjectures are true.