Verification of quantum circuits is essential for guaranteeing correctness of quantum algorithms and/or quantum descriptions across various levels of abstraction. In this work, we show that there are promising ways to check the correctness of quantum circuits using simulative verification and random stimuli. To this end, we investigate how to properly generate stimuli for efficiently checking the correctness of a quantum circuit. More precisely, we introduce, illustrate, and analyze three schemes for quantum stimuli generation---offering a trade-off between the error detection rate (as well as the required number of stimuli) and efficiency. In contrast to the verification in the classical realm, we show (both, theoretically and empirically) that even if only a few randomly-chosen stimuli (generated from the proposed schemes) are considered, high error detection rates can be achieved for quantum circuits. The results of these conceptual and theoretical considerations have also been empirically confirmed---with a grand total of approximately $10^6$ simulations conducted across 50 000 benchmark instances.
We perform formal verification of quantum circuits by integrating several techniques specialized to particular classes of circuits. Our verification methodology is based on the new notion of a reversible miter that allows one to leverage existing techniques for circuit simplification of quantum circuits. For reversible circuits which arise as runtime bottlenecks of key quantum algorithms, we develop several verification techniques and empirically compare them. We also combine existing quantum verification tools with the use of SAT-solvers. Experiments with circuits for Shor's number-factoring …
This short note proposes a symbolic approach for representing and reasoning about quantum circuits using complex, vector or matrix-valued Boolean expressions. A major benefit of this approach is that it allows us to directly borrow the existing techniques and tools for verification of classical logic circuits in reasoning about quantum circuits.
Quantum resource analysis is crucial for designing quantum circuits as well as assessing the viability of arbitrary (error-corrected) quantum computations. To this end, we introduce QUANTIFY, which is an open-source framework for the quantitative analysis of quantum circuits. It is based on Google Cirq and is developed with Clifford+T circuits in mind, and it includes the necessary methods to handle Toffoli+H and more generalised controlled quantum gates, too. Key features of QUANTIFY include: (1) analysis and optimisation methods which are compatible with the surface code, (2) choice between …
Recently much attention has been paid to quantum circuit design to prepare for the future "quantum computation era." Like the conventional logic synthesis, it should be important to verify and analyze the functionalities of generated quantum circuits. For that purpose, we propose an efficient verification method for quantum circuits under a practical restriction. Thanks to the restriction, we can introduce an efficient verification scheme based on decision diagrams called Decision Diagrams for Matrix Functions (DDMFs). Then, we show analytically the advantages of our approach based on DDMFs ov…
We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time dec…
With the race to build large-scale quantum computers and efforts to exploit quantum algorithms for efficient problem solving in science and engineering disciplines, the requirement to have efficient and scalable verification methods are of vital importance. We propose a novel formal verification method that is targeted at Quantum Fourier Transform (QFT) circuits. QFT is a fundamental quantum algorithm that forms the basis of many quantum computing applications. The verification method employs abstractions of quantum gates used in QFT that leads to a reduction of the verification problem from H…
Flag verification techniques are useful in quantum error correction for detecting critical faults. Here we present an application of flag verification techniques to improving post-selected performance of near-term algorithms. We extend the definition of what constitutes a flag by creating error-detection gadgets based on known transformations of unitary operators. In the case of Clifford or near-Clifford circuits, these unitary operators can be chosen to be controlled Pauli gates, leading to gadgets which require only a small number of additional Clifford gates. We show that such flags can imp…
Quantum computations are expressed in general as quantum circuits, which are specified by ordered lists of quantum gates. The resulting specifications are used during the optimisation and execution of the expressed computations. However, the specification format makes it difficult to verify that optimised or executed computations still conform to the initial gate list specifications: showing the computational equivalence between two quantum circuits expressed by different lists of quantum gates is exponentially complex in the worst case. In order to solve this issue, this work presents a deriv…
In this talk, we will describe a framework for assertion-based verification (ABV) of quantum circuits by applying model checking techniques for quantum systems developed in our previous work, in which: (i) Noiseless and noisy quantum circuits are modelled as operator- and super-operator-valued transition systems, respectively, both of which can be further represented by tensor networks. (ii) Quantum assertions are specified by a temporal extension of Birkhoff-von Neumann quantum logic. Their semantics is defined based on the design decision: they will be used in verification of quantum circuit…
We consider the task of verifying the correctness of quantum computation for a restricted class of circuits which contain at most two basis changes. This contains circuits giving rise to the second level of the Fourier Hierarchy, the lowest level for which there is an established quantum advantage. We show that, when the circuit has an outcome with probability at least the inverse of some polynomial in the circuit size, the outcome can be checked in polynomial time with bounded error by a completely classical verifier. This verification procedure is based on random sampling of computational pa…
Practical Log-Depth Quantum State Preparation and Circuit Verification via Tree Tensor Network Compilation
University College London（伦敦大学学院）；Centre for Computational Science, Department of Chemistry, University College London（伦敦大学学院化学系计算科学中心）；Advanced Research Computing Centre, University College London（伦敦大学学院高级计算研究中心）；Informatics Institute, University of Amsterdam（阿姆斯特丹大学信息学研究所）
Quantum computers are reaching a level where interactions between classical and quantum computations can happen in real-time. This marks the advent of a new, broader class of quantum circuits: dynamic quantum circuits. They offer a broader range of available computing primitives that lead to new challenges for design tasks such as simulation, compilation, and verification. Due to the non-unitary nature of dynamic circuit primitives, most existing techniques and tools for these tasks are no longer applicable in an out-of-the-box fashion. In this work, we discuss the resulting consequences for q…
Diners dilemma is one of the most interesting problems in both economic and game theories. Here, we solve this problem for n (number of players) =4 with quantum rules and we are able to remove the dilemma of diners between the Pareto optimal and Nash equilibrium points of the game. We find the quantum strategy that gives maximum payoff for each diner without affecting the payoff and strategy of others. We use the quantum principles of superposition and entanglement that gives supremacy over any classical strategies. We present the circuit implementation for the game, design it on the IBM quant…
We give a scheme for interpreting shaded tangles as quantum circuits, with the property that if two shaded tangles are ambient isotopic, their corresponding computational effects are identical. We analyze 11 known quantum procedures in this way -- including entanglement manipulation, error correction and teleportation -- and in each case present a fully-topological formal verification, yielding generalized procedures in some cases. We also use our methods to identify 2 new procedures, for topological state transfer and quantum error correction. Our formalism yields in some cases significant ne…
State-of-the-art noisy intermediate-scale quantum computers require low-complexity techniques for the mitigation of computational errors inflicted by quantum decoherence. Symmetry verification constitutes a class of quantum error mitigation (QEM) techniques, which distinguishes erroneous computational results from the correct ones by exploiting the intrinsic symmetry of the computational tasks themselves. Inspired by the benefits of quantum switch in the quantum communication theory, we propose beneficial techniques for circuit-oriented symmetry verification that are capable of verifying the c…
Decision problems are the problems whose answer is either YES or NO. As the quantum analogue of $\mathsf{NP}$ (nondeterministic polynomial time), the class $\mathsf{QMA}$ (quantum Merlin-Arthur) contains the decision problems whose YES instance can be verified efficiently with a quantum computer. The problem of deciding the group non-membership (GNM) of a group element is known to be in $\mathsf{QMA}$. Previous works on the verification of GNM required a quantum circuit with $O(n^5)$ group oracle calls. Here we propose an efficient way to verify GNM problems, reducing the circuit depth to $O(1…
The computational advantage of noisy quantum computers has been demonstrated by sampling the bitstrings of quantum random circuits. An important issue is how the performance of quantum devices could be quantified in the so-called "supremacy regime". The standard approach is through the linear cross entropy benchmark (XEB), where the theoretical value of the probability is required for each bitstring. However, the computational cost of XEB grows exponentially. So far, random circuits of the 53-qubit Sycamore chip were verified up to 10 cycles of gates only; the XEB fidelities of deeper circuits…
We present a space-efficient implementation of the quantum verification of matrix products (QVMP) algorithm and demonstrate its functionality by running it on the Aer simulator with two simulation methods: statevector and matrix product state (MPS). We report circuit metrics (gate count, qubit count, circuit depth), transpilation time, simulation time, and a proof of Grover oracle correctness. Our study concludes that while QVMP can be simulated on moderately sized inputs, it cannot scale to a degree where we can observe any quantum advantage on current quantum hardware due to circuit depth an…
We present CertiQ, a verification framework for writing and verifying compiler passes of Qiskit, the most widely-used quantum compiler. To our knowledge, CertiQ is the first effort enabling the verification of real-world quantum compiler passes in a mostly-automated manner. Compiler passes written in the CertiQ interface with annotations can be used to generate verification conditions, as well as the executable code that can be integrated into Qiskit. CertiQ introduces the quantum circuit calculus to enable the efficient checking of equivalence of quantum circuits by encoding such a checking p…
Generalised Bit-Vector Abstractions for Formal Verification of Quantum Error-Detection and Entanglement Circuits over {H,X,C-NOT}: CSS Constructions, Soundness, and Mutation-Based Validation
We present a formulation of quantum circuits where the focus is set on whether a given circuit (made of unitary operators and projective measurements with definite outcomes) does reflect an actually realizable physical experiment. In order to do this, we introduce verifications statements which are purely epistemic assertions indicating whether a outcome is possible at some point and develop our formalism which, in the end, consists in a set of logical rules about verification statements. Finally, we argue that our formalism provides a Lorentz-invariant realistic formulation of quantum circuit…
This paper presents Giallar, a fully-automated verification toolkit for quantum compilers. Giallar requires no manual specifications, invariants, or proofs, and can automatically verify that a compiler pass preserves the semantics of quantum circuits. To deal with unbounded loops in quantum compilers, Giallar abstracts three loop templates, whose loop invariants can be automatically inferred. To efficiently check the equivalence of arbitrary input and output circuits that have complicated matrix semantics representation, Giallar introduces a symbolic representation for quantum circuits and a s…
The development of practical methods for synthesis and verification of complex photonic circuits presents a grand challenge for the nascent field of quantum engineering. Of course, classical electrical engineering provides essential foundations and serves to illustrate the degree of sophistication that can be achieved in automated circuit design. In this paper we explore the utility of term rewriting approaches to the transformation of quantum circuit models, specifically applying rewrite rules for both reduction/verification and robustness analysis of photonic circuits for autonomous quantum …
Quantum computers are on the brink of surpassing the capabilities of even the most powerful classical computers. This naturally raises the question of how one can trust the results of a quantum computer when they cannot be compared to classical simulation. Here we present a verification technique that exploits the principles of measurement-based quantum computation to link quantum circuits of different input size, depth, and structure. Our approach enables consistency checks of quantum computations within a device, as well as between independent devices. We showcase our protocol by applying it…
We introduce a private quantum money scheme with the note verification procedure based on Sampling Matching, a problem in the one-way communication complexity model introduced by Kumar et al.[Nature Communications 10, Article number: 4152]. Our scheme involves a Bank that produces and distributes quantum notes, noteholders who are untrusted and trusted local verifiers of the Bank to whom the holders send their notes in order to carry out transactions. The key aspects of our money scheme include: note verification procedure requiring a single round classical interaction between the local verifi…
Verification is a task to check whether a given quantum state is close to an ideal state or not. In this paper, we show that a variety of many-qubit quantum states can be verified with only sequential single-qubit measurements of Pauli operators. First, we introduce a protocol for verifying ground states of Hamiltonians. We next explain how to verify quantum states generated by a certain class of quantum circuits. We finally propose an adaptive test of stabilizers that enables the verification of all polynomial-time-generated hypergraph states, which include output states of the Bremner-Montan…
Quantum processes, such as quantum circuits, quantum memories, and quantum channels, are essential ingredients in almost all quantum information processing tasks. However, the characterization of these processes remains a daunting task due to the exponentially increasing amount of resources required by traditional methods. Here, by first proposing the concept of quantum process verification, we establish two efficient and practical protocols for verifying quantum processes which can provide an exponential improvement over the standard quantum process tomography and a quadratic improvement over…
We present a formalism that captures the process of proving quantum superiority to skeptics as an interactive game between two agents, supervised by a referee. Bob, is sampling from a classical distribution on a quantum device that is supposed to demonstrate a quantum advantage. The other player, the skeptical Alice, is then allowed to propose mock distributions supposed to reproduce Bob's device's statistics. He then needs to provide witness functions to prove that Alice's proposed mock distributions cannot properly approximate his device. Within this framework, we establish three results. Fi…
While quantum speed-up in solving certain decision problems by a fault-tolerant universal quantum computer has been promised, a timely research interest includes how far one can reduce the resource requirement to demonstrate a provable advantage in quantum devices without demanding quantum error correction, which is crucial for prolonging the coherence time of qubits. We propose a model device made of locally-interacting multiple qubits, designed such that simultaneous single-qubit measurements on it can output probability distributions whose average-case sampling is classically intractable, u…
Mitigating errors by quantum verification and post-selection
School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；Quandela SAS（Quandela公司）；Laboratoire d’Informatique de Paris 6, Centre National de la Recherche Scientifique, Sorbonne Université（巴黎第六大学计算机实验室，法国国家科学研究中心，索邦大学）
We present two protocols for classical verification of quantum depth. Our protocols allow a purely classical verifier to distinguish devices with different quantum circuit depths even in the presence of classical computation. We show that a device with quantum circuit depth at most d will be rejected by the verifier even if the prover applies additional polynomial-time classical computation to cheat. On the other hand, the verifier accepts a device which has quantum circuit depth d' for some d'>d. In our first protocol, we introduce an additional untrusted quantum machine which shares entangle…
Recently, there has been an emergence of useful applications for noisy intermediate-scale quantum (NISQ) devices notably, though not exclusively, in the fields of quantum machine learning and variational quantum algorithms. In such applications, circuits of various depths and composed of different sets of gates are run on NISQ devices. Therefore, it is crucial to find practical ways to capture the general performance of circuits on these devices. Motivated by this pressing need, we modified the standard Clifford randomized benchmarking (RB) and interleaved RB schemes targeting them to hardware…
Experimental verification of Threshold Quantum State Tomography on a fully-reconfigurable photonic integrated circuit
Sapienza Università di Roma（罗马第一大学）；Università di Pavia（帕维亚大学）；European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*, Fondazione Bruno Kessler)（核物理及相关领域欧洲理论研究中心（ECT*，布鲁诺·凯斯勒基金会））；Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche (IFN-CNR)（意大利国家研究委员会光子学与纳米技术研究所）；Politecnico di Milano（米兰理工大学）
Verification of NISQ era quantum devices demands fast classical simulation of large noisy quantum circuits. We present an algorithm based on the stabilizer formalism that can efficiently simulate noisy stabilizer circuits. Additionally, the protocol can efficiently simulate a large set of multi-qubit mixed states that are not mixtures of stabilizer states. The existence of these 'bound states' was previously only known for odd-dimensional systems like qutrits. The algorithm also has the favorable property that circuits with depolarizing noise are simulated much faster than unitary circuits. Th…
Quantum computers promise to efficiently solve important problems classical computers never will. However, in order to capitalize on these prospects, a fully automated quantum software stack needs to be developed. This involves a multitude of complex tasks from the classical simulation of quantum circuits, over their compilation to specific devices, to the verification of the circuits to be executed as well as the obtained results. All of these tasks are highly non-trivial and necessitate efficient data structures to tackle the inherent complexity. Starting from rather straight-forward arrays …
As the width and depth of quantum circuits implemented by state-of-the-art quantum processors rapidly increase, circuit analysis and assessment via classical simulation are becoming unfeasible. It is crucial, therefore, to develop new methods to identify significant error sources in large and complex quantum circuits. In this work, we present a technique that pinpoints the sections of a quantum circuit that affect the circuit output the most and thus helps to identify the most significant sources of error. The technique requires no classical verification of the circuit output and is thus a sca…
Hardy's paradox (equivalently, Hardy's non-locality or Hardy's test) [\href{https://link.aps.org/doi/10.1103/PhysRevLett.68.2981}{L. Hardy, Phys. Rev. Lett. \textbf{68}, 2981 (1992)}] is used to show non-locality without inequalities and it has been tested several times using optical circuits. We, for the first time, experimentally test Hardy's paradox of non-locality in superconducting qubits. For practical verification of Hardy's paradox, we argue that the error-modeling used in optical circuits is not useful for superconducting qubits. So, we propose a new error-modeling for Hardy's paradox…
In this paper, we introduce VeriQBench -- an open source benchmark for quantum circuits. It offers high-level quantum circuit abstractions of various circuit types, including 1) combinational, 2) dynamic, 3) sequential, and 4) variational quantum circuits, which cover almost all existing types of quantum circuits in the literature. Meanwhile, VeriQBench is a versatile benchmark which can be used in verifying quantum software for different applications, as is evidenced by the existing works including quantum circuit verification (e.g., equivalence checking [Hon+21a; WLY21] and model checking [Y…
To guarantee the normal functioning of quantum devices in different scenarios, appropriate benchmarking tool kits are quite significant. Inspired by the recent progress on quantum state verification, here we establish a general framework of verifying a target unitary gate. In both the non-adversarial and adversarial scenarios, we provide efficient methods to evaluate the performance of verification strategies for any qudit unitary gate. Furthermore, we figure out the optimal strategy and its realization with local operations. Specifically, for the commonly-used quantum gates like single qubit …
Quantum computing promises to solve some important problems faster than conventional computations ever could. Currently available NISQ devices on which first practical applications are already executed demonstrate the potential -- with future fault-tolerant quantum hardware for more demanding applications on the horizon. Nonetheless, the advantages in computing power come with challenges to be addressed in the design automation and software development community. In particular, non-quantum representations of states and operations, which provide the basis, e.g., for quantum circuit simulation o…
