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Bulletin of the American Physical Society

Quantum Circuit and Algorithm Validation With Prove-It

Wayne M. Witzel, Kenneth Rudinger, Robert D. Carr, Mohan Sarovar

Published Mar 8, 2019• 0 citations
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Verification of Quantum Circuits

Robert Wille, Lukas Burgholzer

Springer eBooks•Dec 20, 2024•0 citations

Simulating Quantum Circuits with Arbitrary Local Noise Using Pauli Propagation

Armando Angrisani, Antonio Anna Mele, Manuel S. Rudolph +2 more

We present a polynomial-time classical algorithm for estimating expectation values of arbitrary observables on typical quantum circuits under any incoherent local noise, including non-unital or dephasing. Although previous research demonstrated that some carefully designed quantum circuits affected by non-unital noise cannot be efficiently simulated, we show that this does not apply to average-case circuits, as these can be efficiently simulated using Pauli-path methods. Specifically, we prove that, with high probability over the circuit gates’ choice, Pauli propagation algorithms with tailored truncation strategies achieve an inversely polynomially small simulation error. This result holds for arbitrary circuit topologies and for any local noise, under the assumption that the distribution of each circuit layer is invariant under single-qubit random gates. Under the same minimal assumptions, we also prove that most noisy circuits can be truncated to an effective logarithmic depth for the task of estimating expectation values of observables, thus generalizing prior results to a significantly broader class of circuit ensembles. We further numerically validate our algorithm with simulations on a 6 × 6 lattice of qubits under the effects of amplitude damping and dephasing noise, as well as real-time dynamics on an 11 × 11 lattice of qubits affected by amplitude damping.

arXiv:2501.13101•PRX Quantum•Mar 17, 2026•8 citations

Verification of Quantum Circuits

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Springer eBooks•Jan 1, 2022•5 citations

Big Step: A fast quantum algorithm for nonce discovery in Proof-of-Work blockchains

Young-Hoon Park

Since the emergence of quantum computing, Grover’s algorithm has been widely studied as a method to attack Proof-of-Work (PoW) blockchains. However, it requires multiple oracle and diffuser iterations, causing high computational overhead and ancilla qubit usage. In this paper, we propose a novel quantum algorithm that efficiently finds valid nonces in PoW-based blockchains without iterative amplification. It introduces a non-unitary quantum operator that directly generates a superposition of valid nonces in one step. We also present an efficient quantum circuit implementation and theoretically prove that our algorithm significantly reduces computational resources compared to conventional Grover-based methods.

ICT Express•Dec 2, 2025•2 citations

Efficient equivalence checking of Clifford-U circuits with shared single-qubit unitaries

Daisuke Sakamoto, Soshun Naito, Yusei Mori +1 more

Quantum circuit equivalence checking asks whether two circuits implement the same unitary. It guarantees compiler correctness and safe optimization, yet most existing approaches scale exponentially with the number of qubits or the circuit depth, or are restricted to specific circuit structures. In this work, we present an equivalence-checking method for circuits formed by arbitrary single-qubit layers interleaved with Clifford layers. This pattern is common in variational quantum algorithms and Hamiltonian simulation via Trotter decomposition. It can also represent any unitary with sufficient depth. We prove the existence of an efficient classical algorithm that determines whether a pair of circuits with shared single-qubit layers are equivalent for every possible choice of the shared single-qubit unitaries. The same algorithm can also certify their non-equivalence for fixed assignments of single-qubit unitaries. Our framework supports the validation of emerging quantum compilers and facilitate the discovery of novel circuit optimization passes.

quant-pharXiv:2603.12697•Mar 16, 2026•0 citations

Near-perfect Reachability of Variational Quantum Search with Depth-1 Ansatz

Junpeng Zhan

Grover's search algorithm is renowned for its dramatic speedup in solving many important scientific problems. The recently proposed Variational Quantum Search (VQS) algorithm has shown an exponential advantage over Grover's algorithm for up to 26 qubits. However, its advantage for larger numbers of qubits has not yet been proven. Here we show that the exponentially deep circuit required by Grover's algorithm can be replaced by a multi-controlled NOT gate together with either a single layer of Ry gates or two layers of circuits consisting of Hadamard and NOT gates, which is valid for any number of qubits greater than five. We prove that the VQS, with a single layer of Ry gates as its Ansatz, has near-perfect reachability in finding the good element of an arbitrarily large unstructured data set, and its reachability exponentially improves with the number of qubits, where the reachability is defined to quantify the ability of a given Ansatz to generate an optimal quantum state. Numerical studies further validate the excellent reachability of the VQS. Proving the near-perfect reachability of the VQS, with a depth-1 Ansatz, for any number of qubits completes an essential step in proving its exponential advantage over Grover's algorithm for any number of qubits, and the latter proving is significant as it means that the VQS can efficiently solve NP-complete problems.

arXiv:2301.13224•Jan 30, 2023•1 citation
We present a polynomial-time classical algorithm for estimating expectation values of arbitrary observables on typical quantum circuits under any incoherent local noise, including non-unital or dephasing. Although previous research demonstrated that some carefully designed quantum circuits affected by non-unital noise cannot be efficiently simulated, we show that this does not apply to average-case circuits, as these can be efficiently simulated using Pauli-path methods. Specifically, we prove that, with high probability over the circuit gates’ choice, Pauli propagation algorithms with tailored truncation strategies achieve an inversely polynomially small simulation error. This result holds for arbitrary circuit topologies and for any local noise, under the assumption that the distribution of each circuit layer is invariant under single-qubit random gates. Under the same minimal assumptions, we also prove that most noisy circuits can be truncated to an effective logarithmic depth for the task of estimating expectation values of observables, thus generalizing prior results to a significantly broader class of circuit ensembles. We further numerically validate our algorithm with simulations on a 6 × 6 lattice of qubits under the effects of amplitude damping and dephasing noise, as well as real-time dynamics on an 11 × 11 lattice of qubits affected by amplitude damping.
Since the emergence of quantum computing, Grover’s algorithm has been widely studied as a method to attack Proof-of-Work (PoW) blockchains. However, it requires multiple oracle and diffuser iterations, causing high computational overhead and ancilla qubit usage. In this paper, we propose a novel quantum algorithm that efficiently finds valid nonces in PoW-based blockchains without iterative amplification. It introduces a non-unitary quantum operator that directly generates a superposition of valid nonces in one step. We also present an efficient quantum circuit implementation and theoretically prove that our algorithm significantly reduces computational resources compared to conventional Grover-based methods.
Quantum circuit equivalence checking asks whether two circuits implement the same unitary. It guarantees compiler correctness and safe optimization, yet most existing approaches scale exponentially with the number of qubits or the circuit depth, or are restricted to specific circuit structures. In this work, we present an equivalence-checking method for circuits formed by arbitrary single-qubit layers interleaved with Clifford layers. This pattern is common in variational quantum algorithms and Hamiltonian simulation via Trotter decomposition. It can also represent any unitary with sufficient depth. We prove the existence of an efficient classical algorithm that determines whether a pair of circuits with shared single-qubit layers are equivalent for every possible choice of the shared single-qubit unitaries. The same algorithm can also certify their non-equivalence for fixed assignments of single-qubit unitaries. Our framework supports the validation of emerging quantum compilers and facilitate the discovery of novel circuit optimization passes.
Grover's search algorithm is renowned for its dramatic speedup in solving many important scientific problems. The recently proposed Variational Quantum Search (VQS) algorithm has shown an exponential advantage over Grover's algorithm for up to 26 qubits. However, its advantage for larger numbers of qubits has not yet been proven. Here we show that the exponentially deep circuit required by Grover's algorithm can be replaced by a multi-controlled NOT gate together with either a single layer of Ry gates or two layers of circuits consisting of Hadamard and NOT gates, which is valid for any number of qubits greater than five. We prove that the VQS, with a single layer of Ry gates as its Ansatz, has near-perfect reachability in finding the good element of an arbitrarily large unstructured data set, and its reachability exponentially improves with the number of qubits, where the reachability is defined to quantify the ability of a given Ansatz to generate an optimal quantum state. Numerical studies further validate the excellent reachability of the VQS. Proving the near-perfect reachability of the VQS, with a depth-1 Ansatz, for any number of qubits completes an essential step in proving its exponential advantage over Grover's algorithm for any number of qubits, and the latter proving is significant as it means that the VQS can efficiently solve NP-complete problems.