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arXiv:1903.08357 · Leibniz international proceedings in informatics

Quantum Relational Hoare Logic with Expectations

Yangjia Li, Dominique Unruh

Published Mar 20, 2019• 20 citations
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Abstract

We present a variant of the quantum relational Hoare logic from (Unruh, POPL 2019) that allows us to use "expectations" in pre- and postconditions. That is, when reasoning about pairs of programs, our logic allows us to quantitatively reason about how much certain pre-/postconditions are satisfied that refer to the relationship between the programs inputs/outputs.

DOI
10.4230/lipics.icalp.2021.136

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We present a variant of the quantum relational Hoare logic from (Unruh, POPL 2019) that allows us to use "expectations" in pre- and postconditions. That is, when reasoning about pairs of programs, our logic allows us to quantitatively reason about how much certain pre-/postconditions are satisfied that refer to the relationship between the programs inputs/outputs.
We derive a variant of quantum Hoare logic (QHL), called applied quantum Hoare logic (aQHL for short), by: 1. restricting QHL to a special class of preconditions and postconditions, namely projections, which can significantly simplify verification of quantum programs and are much more convenient when used in debugging and testing; and 2. adding several rules for reasoning about robustness of quantum programs, i.e. error bounds of outputs. The effectiveness of aQHL is shown by its applications to verify two sophisticated quantum algorithms: HHL (Harrow-Hassidim-Lloyd) for solving systems of linear equations and qPCA (quantum Principal Component Analysis).
We present a logic for reasoning about pairs of interactive quantum programs – quantum relational Hoare logic (qRHL). This logic follows the spirit of probabilistic relational Hoare logic (Barthe et al. 2009) and allows us to formulate how the outputs of two quantum programs relate given the relationship of their inputs. Probabilistic RHL was used extensively for computer-verified security proofs of classical cryptographic protocols. Since pRHL is not suitable for analyzing quantum cryptography, we present qRHL as a replacement, suitable for the security analysis of post-quantum cryptography and quantum protocols. The design of qRHL poses some challenges unique to the quantum setting, e.g., the definition of equality on quantum registers. Finally, we implemented a tool for verifying proofs in qRHL and developed several example security proofs in it.
Quantum Hoare logic allows us to reason about quantum programs. We present an extension of quantum Hoare logic that introduces “ghost variables” to extend the expressive power of pre-/postconditions. Ghost variables are variables that do not actually occur in the program and are allowed to have arbitrary quantum states (in a sense, they are existentially quantified), and be entangled with program variables. Ghost variables allow us to express properties such as the distribution of a program variable or the fact that a variable has classical content. And as a case study, we show how quantum Hoare logic with ghost variables can be used to prove the security of the quantum one-time pad.
Hoare logic is a foundation of axiomatic semantics of classical programs and it provides effective proof techniques for reasoning about correctness of classical programs. To offer similar techniques for quantum program verification and to build a logical foundation of programming methodology for quantum computers, we develop a full-fledged Hoare logic for both partial and total correctness of quantum programs. It is proved that this logic is (relatively) complete by exploiting the power of weakest preconditions and weakest liberal preconditions for quantum programs.