The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

Vilmart, Renaud

doi:10.1007/978-3-030-71995-1_27

Cited by 12 publications

(39 citation statements)

References 13 publications

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“…We begin by briefly reviewing the theory of path sums [2,31]. A path sum representation of a linear operator Ψ : C 2 m → C 2 n is an expression for Ψ as a sum indexed by binary variables such as…”

Section: Path Sumsmentioning

confidence: 99%

“…The sum-over-paths representations of linear operators has been studied extensively in the context of quantum information [13,9,27,24,19,14]. Recent work on the connection to graphical calculi [20,31] has shown that path sums form a universal model for linear operators over 2 n -dimensional Hilbert spaces through direct translations from universal graphical calculi such as the ZH-calculus [7].…”

Section: Path Sumsmentioning

confidence: 99%

“…which act, respectively, as the unit and counit in the category FdHilb [31], can be written as the following path sums:…”

Section: Path Sumsmentioning

confidence: 99%

“…Equational reasoning A major utility of the path sum representation [2] comes from the ability to perform equational reasoning. Complete equational theories of Clifford unitaries [2] and more general stabilizer operations [31] have previously been developed. We reformulate these theories here using locally free variables to simplify their presentation.…”

Section: Path Sumsmentioning

confidence: 99%

“…Equations ( 2), (3) and ( 4) are restatements of [2, Proposition 3.1]. Equation ( 5) is a generalization of the (ket) rule given in [31,Figure 3] and can be derived from Equation (3) as follows:…”

Section: Path Sumsmentioning

confidence: 99%

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Symbolic synthesis of Clifford circuits and beyond

Amy¹,

Bennett-Gibbs²,

Ross³

2022

Preprint

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Path sums are a convenient symbolic formalism for quantum operations with applications to the simulation, optimization, and verification of quantum protocols. Unlike quantum circuits, path sums are not limited to unitary operations, but can express arbitrary linear ones. Two problems, therefore, naturally arise in the study of path sums: the unitarity problem and the extraction problem. The former is the problem of deciding whether a given path sum represents a unitary operator. The latter is the problem of constructing a quantum circuit, given a path sum promised to represent a unitary operator.In this paper, we show that the unitarity problem is co-NP-hard in general, but that it is in P when restricted to Clifford path sums. We then provide an algorithm to synthesize a Clifford circuit from a unitary Clifford path sum. The circuits produced by our extraction algorithm are of the form C1HC2, where C1 and C2 are Hadamard-free circuits and H is a layer of Hadamard gates. We also provide a heuristic generalization of our extraction algorithm to arbitrary path sums. While this algorithm is not guaranteed to succeed, it often succeeds and typically produces natural looking circuits. Alongside applications to the optimization and decompilation of quantum circuits, we demonstrate the capability of our algorithm by synthesizing the standard quantum Fourier transform directly from a path sum.

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Section: Path Sumsmentioning

confidence: 99%

Section: Path Sumsmentioning

confidence: 99%

“…which act, respectively, as the unit and counit in the category FdHilb [31], can be written as the following path sums:…”

Section: Path Sumsmentioning

confidence: 99%

Section: Path Sumsmentioning

confidence: 99%

Section: Path Sumsmentioning

confidence: 99%

See 3 more Smart Citations

Symbolic synthesis of Clifford circuits and beyond

Amy¹,

Bennett-Gibbs²,

Ross³

2022

Preprint

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The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

Vilmart

2021

Preprint

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The "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems, and is a tool that is used in formal verification of such systems.We give here a new set of rewrite rules for the formalism, and show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics. We show that the rewriting is terminating, but not confluent (which is expected from the universality of the fragment). We do so using the connection between Sum-over-Paths and graphical language ZH-calculus, and also show how the axiomatisation translates into the latter. We provide generalisations of the presented rewrite rules, that can prove useful when trying to reduce terms in practice, and we show how to graphically make sense of these new rules.We show how to enrich the rewrite system to reach completeness for the dyadic fragments of quantum computation, used in particular in the Quantum Fourier Transform, and obtained by adding phase gates with dyadic multiples of π to the Toffoli-Hadamard gate-set.Finally, we show how to perform sums and concatenation of arbitrary terms, something which is not native in a system designed for analysing gate-based quantum computation, but necessary when considering Hamiltonian-based quantum computation.

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Completeness of the ZH-calculus

et al. 2023

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There are various gate sets used for describing quantum computation. A particularly popular one consists of Clifford gates and arbitrary single-qubit phase gates. Computations in this gate set can be elegantly described by the ZX-calculus, a graphical language for a class of string diagrams describing linear maps between qubits. The ZX-calculus has proven useful in a variety of areas of quantum information, but is less suitable for reasoning about operations outside its natural gate set such as multi-linear Boolean operations like the Toffoli gate. In this paper we study the ZH-calculus, an alternative graphical language of string diagrams that does allow straightforward encoding of Toffoli gates and other more complicated Boolean logic circuits. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over Z[12], which correspond to the approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring R where 1+1 is not a zero-divisor.

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The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

Cited by 12 publications

References 13 publications

Symbolic synthesis of Clifford circuits and beyond

Symbolic synthesis of Clifford circuits and beyond

The Structure of Sum-Over-Paths, its Consequences, and Completeness for Clifford

Completeness of the ZH-calculus

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