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 algorithm, containing thousands of gates, show improvements in efficiency by 3-4 orders of magnitude.
Quantum computing promises exponential speed-ups for important simulation and optimization problems. It also poses new CAD problems that are similar to, but more challenging, than the related problems in classical (non-quantum) CAD, such as determining if two states or circuits are functionally equivalent. While differences in classical states are easy to detect, quantum states, which are represented by complex-valued vectors, exhibit subtle differences leading to several notions of equivalence. This provides flexibility in optimizing quantum circuits, but leads to difficult new equivalence-ch…
As state-of-the-art quantum computers are capable of running increasingly complex algorithms, the need for automated methods to design and test potential applications rises. Equivalence checking of quantum circuits is an important, yet hardly automated, task in the development of the quantum software stack. Recently, new methods have been proposed that tackle this problem from widely different perspectives. One of them is based on the ZX-calculus, a graphical rewriting system for quantum computing. However, the power and capability of this equivalence checking method has barely been explored. …
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 Due to the non-unitary nature of dynamic circuit primitives, most existing techniques and tools for these tasks are no longer applicable in an out-of-the-box fashion.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…
In this paper, we introduce the model of quantum Mealy machines and study the equivalence checking and minimisation problems of them. Two efficient algorithms are developed for checking equivalence of two states in the same machine and for checking equivalence of two machines. As an application, they are used in equivalence checking of quantum circuits. Moreover, the minimisation problem is proved to be in $\textbf{PSPACE}$.
Quantum computing will change the way we tackle certain problems. It promises to dramatically speed-up many chemical, financial, and machine-learning applications. However, to capitalize on those promises, complex design flows composed of steps such as compilation, decomposition, or mapping need to be employed before being able to execute a conceptual quantum algorithm on an actual device. This results in descriptions at various levels of abstraction which may significantly differ from each other. The complexity of the underlying design problems necessitates to not only provide efficient solut…
We define a formal framework for equivalence checking of sequential quantum circuits. The model we adopt is a quantum state machine, which is a natural quantum generalisation of Mealy machines. A major difficulty in checking quantum circuits (but not present in checking classical circuits) is that the state spaces of quantum circuits are continuums. This difficulty is resolved by our main theorem showing that equivalence checking of two quantum Mealy machines can be done with input sequences that are taken from some chosen basis (which are finite) and have a length quadratic in the dimensions …
Despite the rapid development of quantum computing these years, state-of-the-art quantum devices still contain only a very limited number of qubits. One possible way to execute more realistic algorithms in near-term quantum devices is to employ dynamic quantum circuits, in which measurements can happen during the circuit and their outcomes are used to control other parts of the circuit. This technique can help to significantly reduce the resources required to achieve a given accuracy of a quantum algorithm. However, since this type of quantum circuits are more flexible, their verification is m…
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…
Suppose two quantum circuit chips are located at different places, for which we do not have any prior knowledge, and cannot see the internal structures either. If we want to find out whether they have the same functions or not with certainty, what should we do? In this paper, we show that this realistic problem can be solved completely from the viewpoints of quantum nonlocality. Specifically, we design an elegant protocol that examines underlying quantum nonlocality, where the strongest nonlocality can be observed if and only if two quantum circuits are equivalent to each other. We show that t…
Variational quantum algorithms have been introduced as a promising class of quantum-classical hybrid algorithms that can already be used with the noisy quantum computing hardware available today by employing parameterized quantum circuits. Considering the non-trivial nature of quantum circuit compilation and the subtleties of quantum computing, it is essential to verify that these parameterized circuits have been compiled correctly. Established equivalence checking procedures that handle parameter-free circuits already exist. However, no methodology capable of handling circuits with parameters…
We study the fundamental design automation problem of equivalence checking in the NISQ (Noisy Intermediate-Scale Quantum) computing realm where quantum noise is present inevitably. The notion of approximate equivalence of (possibly noisy) quantum circuits is defined based on the Jamiolkowski fidelity which measures the average distance between output states of two super-operators when the input is chosen at random. By employing tensor network contraction, we present two algorithms, aiming at different situations where the number of noises varies, for computing the fidelity between an ideal qua…
In this paper, we study a model of quantum Markov chains that is a quantum analogue of Markov chains and is obtained by replacing probabilities in transition matrices with quantum operations. We show that this model is very suited to describe hybrid systems that consist of a quantum component and a classical one, although it has the same expressive power as another quantum Markov model proposed in the literature. Indeed, hybrid systems are often encountered in quantum information processing; for example, both quantum programs and quantum protocols can be regarded as hybrid systems. Thus, we fu…
It is well-known that deciding equivalence of logic circuits is a coNP-complete problem. As a corollary, the problem of deciding weak equivalence of reversible circuits, i.e. ignoring the ancilla bits, is also coNP-complete. The complexity of deciding strong equivalence, including the ancilla bits, is less obvious and may depend on gate set. Here we use Barrington's theorem to show that deciding strong equivalence of reversible circuits built from the Fredkin gate is coNP-complete. This implies coNP-completeness of deciding strong equivalence for other commonly used universal reversible gate s…
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…
We prove the equivalence between adiabatic quantum computation and quantum computation in the circuit model. An explicit adiabatic computation procedure is given that generates a ground state from which the answer can be extracted. The amount of time needed is evaluated by computing the gap. We show that the procedure is computationally efficient.
We present a rigorous proof that quantum circuit algorithm can be transformed into quantum adiabatic algorithm with the exact same time complexity. This means that from a quantum circuit algorithm of $L$ gates we can construct a quantum adiabatic algorithm with time complexity of $O(L)$. Additionally, our construction shows that one may exponentially speed up some quantum adiabatic algorithms by properly choosing an evolution path.
Realizing a conceptual quantum algorithm on an actual physical device necessitates the algorithm's quantum circuit description to undergo certain transformations in order to adhere to all constraints imposed by the hardware. In this regard, the individual high-level circuit components are first synthesized to the supported low-level gate-set of the quantum computer, before being mapped to the target's architecture---utilizing several optimizations in order to improve the compilation result. Specialized tools for this complex task exist, e.g., IBM's Qiskit, Google's Cirq, Microsoft's QDK, or Ri…
Atomic (qubit) and optical or microwave (modal) phase-estimation protocols are placed on the same footing in terms of quantum-circuit diagrams. Circuit equivalences are used to demonstrate the equivalence of protocols that achieve the Heisenberg limit by employing entangled superpositions of Fock states, such as N00N states. The key equivalences are those that disentangle a circuit so that phase information is written exclusively on a mode or modes or on a qubit. The Fock-state-superposition phase-estimation circuits are converted to use entangled coherent-state superpositions; the resulting p…
Nielsen \cite{Nielsen05} recently asked the following question: "What is the minimal size quantum circuit required to exactly implement a specified $% \mathit{n}$-qubit unitary operation $U$, without the use of ancilla qubits?" Nielsen was able to prove that a lower bound on the minimal size circuit is provided by the length of the geodesic between the identity $I$ and $U$, where the length is defined by a suitable Finsler metric on $SU(2^{n})$. We prove that the minimum circuit size that simulates $U$ is in linear relation with the geodesic length and simulation parameters, for the given Fins…
In order to establish the computational equivalence between quantum Turing machines (QTMs) and quantum circuit families (QCFs) using Yao's quantum circuit simulation of QTMs, we previously introduced the class of uniform QCFs based on an infinite set of elementary gates, which has been shown to be computationally equivalent to the polynomial-time QTMs (with appropriate restriction of amplitudes) up to bounded error simulation. This result implies that the complexity class BQP introduced by Bernstein and Vazirani for QTMs equals its counterpart for uniform QCFs. However, the complexity classes …
An equivalence between the $\mathrm{Schr\ddot{o}dinger}$ dynamics of a quantum system with a finite number of basis states and a classical dynamics is presented. The equivalence is an isomorphism that connects in univocal way both dynamical systems. We treat the particular case of neutral kaons and found a class of electric networks uniquely related to the kaon system finding the complete map between the matrix elements of the effective Hamiltonian of kaons and those elements of the classical dynamics of the networks. As a consequence, the relevant $ε$ parameter that measures CP violation in t…
Università degli Studi di Padova（帕多瓦大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；Padua Quantum Technologies Research Center（帕多瓦量子技术中心）
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…
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…
We prove that a quantum circuit together with measurement apparatuses and EPR sources can be fully verified without any reference to some other trusted set of quantum devices. Our main assumption is that the physical system we are working with consists of several identifiable sub-systems, on which we can apply some given gates locally. To achieve our goal we define the notions of simulation and equivalence. The concept of simulation refers to producing the correct probabilities when measuring physical systems. To enable the efficient testing of the composition of quantum operations, we introdu…
Quantum computing promises remarkable approaches for processing information, but new tools are needed to compile program representations into the physical instructions required by a quantum computer. Here we present a novel adaptation of the multi-level intermediate representation (MLIR) integrated into a quantum compiler that may be used for checking program execution. We first present how MLIR enables quantum circuit transformations for efficient execution on quantum computing devices and then give an example of compiler transformations based on so-called mirror circuits. We demonstrate that…
