Formally Verifying Quantum Phase Estimation Circuits with 1,000+ Qubits

Jul 16, 2026

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Scalable formal verification methodology for Quantum Phase Estimation (QPE) circuits maps quantum circuit functional behavior from Hilbert space into a bit-vector domain. The method builds a symbolic qubit abstraction using quantifier-free bitvector logic, modeling superposition, rotation, and measurement. Correctness properties aligned with the abstraction are specified and proven sound via formal proofs. It efficiently verifies QPE circuits with up to 6 precision qubits and 1,024 phase qubits within under 7.5 GB memory.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages6

Reading Time15 min

Introduction

Background

Existing Approaches

Formally Verifying Quantum Phase Estimation Circuits with 1,000+ Qubits Arun Govindankutty and Sudarshan K. Srinivasan Electrical and Computer Engineering Department, North Dakota State University, Fargo, ND, USA arXiv :2603 .08762v3 [ quant-ph] 19 Mar 2026 Abstract—We present a scalable formal verification methodology for Quantum Phase Estimation (QPE) circuits. Our approach uses a symbolic qubit abstraction based on quantifier-free bitvector logic, capturing key quantum phenomena, including superposition, rotation, and measurement. The proposed methodology maps quantum circuit functional behaviour from Hilbert space toa bit-vector domain. We develop formal properties aligned with this abstraction to ensure functional correctness of QPE circuits. The method scales efficiently, verifying QPE circuits with up to 6 precision qubits and 1,024 phase qubits using under 7.5 GB of memory. Index Terms—formal verification, quantum computing, quantum circuit verification, quantum phase estimation. I. INTRODUCTION Quantum computing has demonstrated the potential to surpass classical computation in several classes of problems across domains such as communication, machine learning, materials science, medicine, and optimization, enabling progress toward quantum advantage [1]–[3] . Central to many of these advances are quantum algorithms that address problems intractable for classical systems. Among them, Quantum Phase Estimation (QPE) is a core primitive, employed in key applications including Shor’s algorithm [4], quantum counting and amplitude amplification [5], eigenvalue estimation [6], and quantum linear system solvers [7] . Realizing these applications in practice requires QPE circuits to scale to thousands of qubits. However, increasing circuit scale also amplifies design and functional complexity, significantly raising the risk of implementation errors. Therefore, establishing the functional correctness of large-scale QPE circuits is a critical requirement for reliable quantum computation. Quantum circuits operate over complex Hilbert spaces and exhibit uniquely quantum effects such as superposition, entanglement, and measurement, making functional verification inherently challenging. Formal verification is well suited to uncover subtle design errors while ensuring design correctness [8] . Recent work has extended these techniques to the quantum domain by adapting classical formal methods to reason about quantum program behavior [9], [10] . This work explicitly incorporates measurement into the abstraction framework to the work presented in [11], yielding a generic symbolic qubit abstraction based on quantifier free bit-vector logic. The resulting model enables functional correctness verification using Satisfiability Modulo Theories (SMT) solvers. Our main contributions are as follows: • A unified qubit abstraction integrating rotational, superposition, and measurement operations. • Symbolic abstract modelling of controlled modular exponentiation in QPE circuits. • Formal specification of correctness properties capturing QPE functionality. • Soundness lemmas and formal proofs of the proposed verification methodology. The required theoretical background is presented next. II. BACKGROUND This section briefly reviews qubits, quantum gates, and Quantum Phase Estimation (QPE) . Detailed treatments can be found in [12]–[14] . The basic unit of quantum information is the qubit, represented as a linear combination of the computational basis states |0⟩ and |1⟩, defined as |0⟩ = 􀀔10􀀕 , |1⟩ = 􀀔01􀀕 . (1) Fig. 1. (a) Quantum phase estimation circuit. (b)3-qubit inverse quantum Fourier transform circuit [11] . (c) Quantum phase estimation circuit with 3 precision qubits and 1 phase qubit implemented in IBM-Qiskit. Quantum gates are unitary operators that act on qubits to transform their states. Given a unitary operator U, an eigenstate |ψ⟩, and n precision (ancilla) qubits, a QPE circuit estimates the phase θ

FAQs

What verification approach is proposed for Quantum Phase Estimation (QPE) circuits?

A scalable formal verification methodology maps QPE circuit functional behavior from Hilbert space to a bit-vector domain using a symbolic qubit abstraction and quantifier-free bitvector logic.

Which quantum phenomena are captured by the symbolic qubit abstraction?

The abstraction models key quantum behaviors including superposition, rotation, and measurement, enabling correctness verification against formal properties.

What circuit sizes can the methodology verify efficiently?

The method verifies QPE circuits with up to 6 precision qubits and 1,024 phase qubits while using under 7.5 GB of memory.

Jul 16, 2026

Upload by Jiven