A verified optimizer for Quantum circuits

Abstract

We present VOQC, the first fully verified optimizer for quantum circuits, written using the Coq proof assistant. Quantum circuits are expressed as programs in a simple, low-level language called SQIR, a simple quantum intermediate representation, which is deeply embedded in Coq. Optimizations and other transformations are expressed as Coq functions, which are proved correct with respect to a semantics of SQIR programs. SQIR uses a semantics of matrices of complex numbers, which is the standard for quantum computation, but treats matrices symbolically in order to reason about programs that use an arbitrary number of quantum bits. SQIR's careful design and our provided automation make it possible to write and verify a broad range of optimizations in VOQC, including full-circuit transformations from cutting-edge optimizers.

Verified tensor-program optimization via high-level scheduling rewrites

“…Recent work develops rewrite-rule systems for optimizing array programs [Fu et al 2021;Steuwer et al 2015], but they generally treat rewrites as axioms and provide no formal guarantees. Previous work such as the VOQC quantum-circuit optimizer also shows how a tactic engine and interactive theorem prover provide a natural framework for building a verified program-optimization framework [Hietala et al 2021]. However, they use to validate prewritten optimization procedures, while we focus on step-by-step manual derivation of optimizations for specific programs.…”

Pauli Error Propagation-Based Gate Reschedulingfor Quantum Circuit Error Mitigation

“…Complex gates should be constructed using elementary gates supported by the NISQ architecture. To reduce the depth and the gate count of the circuit, several gatelevel optimization techniques are applied including template matching [6] and gate reordering [7] based techniques. In the former one, a cascade of quantum gates is substituted with a sub-circuit with a lower gate count, while in the later one, consecutive gates cancel each other based on the commutation rules of the quantum gates [4], which are defined as follows:…”

EasyPQC: Verifying Post-Quantum Cryptography

“…Besides, a quantum Hoare logic with ghost variables is introduced in [45] which is useful for dealing with mixed states/distributions. Practical tools have also been developed in recent years, including [18,35] in Isabelle/HOL, [27,28,39] in Coq; all of them are based on matrix representation and calculations. They show that deriving and verifying general quantum properties are relatively difficult in implementation (with unexpected times and length of codes), for example, verifying Grover's search algorithm needs around 770 lines of code and one-person week in [27] and over 3000 lines in [35].…”