Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus

Lemonnier, Louis; Wetering, John van de; Kissinger, Aleks

doi:10.4204/eptcs.340.10

Cited by 12 publications

(22 citation statements)

References 24 publications

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“…The extraction method in this paper maps a measurement pattern/ZX-diagram directly into the form of a PDDAG, allowing us to formally demonstrate that each of the graph-theoretic rewrites can be simulated by moving Clifford-angled rotations through the PDDAG. This complements existing work on deriving equivalences between the PathSum formalism and the ZH calculus [3,4,23] and between different graphical calculi [22,28] which have shown to provide useful insight on the semantics of operations for the more abstract calculi.…”

Section: Introductionsupporting

confidence: 62%

Relating Measurement Patterns to Circuits via Pauli Flow

Simmons¹

2021

Electron. Proc. Theor. Comput. Sci.

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The one-way model of Measurement-Based Quantum Computing and the gate-based circuit model give two different presentations of how quantum computation can be performed. There are known methods for converting any gate-based quantum circuit into a one-way computation, whereas the reverse is only efficient given some constraints on the structure of the measurement pattern. Causal flow and generalised flow have already been shown as sufficient, with efficient algorithms for identifying these properties and performing the circuit extraction. Pauli flow is a weaker set of conditions that extends generalised flow to use the knowledge that some vertices are measured in a Pauli basis. In this paper, we show that Pauli flow can similarly be identified efficiently and that any measurement pattern whose underlying graph admits a Pauli flow can be efficiently transformed into a gate-based circuit without using ancilla qubits. We then use this relationship to derive simulation results for the effects of graph-theoretic rewrites in the ZX-calculus using a more circuit-like data structure we call the Pauli Dependency DAG. Relating Measurement Patterns to Circuits via Pauli Flow Gate Equivalent Pauli exponentials CX ct e −i π 4 Z c X t e i π 4 Z c e i π 4 X t CZ ct e −i π 4 Z c Z t e i π 4 Z c e i π 4 Z t RZ(α) e −i α 2 Z RX (α) e −i α 2 X H e −i π 4 Z e −i π 4 X e −i π 4 Z CCX abt e −i π 8 Z a Z b X t e i π 8 Z a Z b e i π 8 Z a X t e i π 8 Z b X t e −i π 8 Z a e −i π 8 Z b e −i π 8 X t

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Section: Introductionsupporting

confidence: 62%

Relating Measurement Patterns to Circuits via Pauli Flow

Simmons¹

2021

Electron. Proc. Theor. Comput. Sci.

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“…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%

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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“…Finally, we can mention that the path-sum formalism described in Section 5.2 below can be used in this context. Indeed, it was shown [108,162] that one can pass from one formalism to the other "freely", allowing us to apply strategies for path-sums to the ZX-Calculus.…”

Section: Algorithms and Toolsmentioning

confidence: 99%

Formal Methods for Quantum Programs: A Survey

Chareton¹,

Bardin²,

Lee³

et al. 2021

Preprint

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While recent progress in quantum hardware open the door for significant speedup in certain key areas (cryptography, biology, chemistry, optimization, machine learning, etc), quantum algorithms are still hard to implement right, and the validation of such quantum programs is a challenge. Moreover, importing the testing and debugging practices at use in classical programming is extremely difficult in the quantum case, due to the destructive aspect of quantum measurement. As an alternative strategy, formal methods are prone to play a decisive role in the emerging field of quantum software. Recent works initiate solutions for problems occurring at every stage of the development process: high-level program design, implementation, compilation, etc. We review the induced challenges for an efficient use of formal methods in quantum computing and the current most promising research directions.

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Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus

Cited by 12 publications

References 24 publications

Relating Measurement Patterns to Circuits via Pauli Flow

Relating Measurement Patterns to Circuits via Pauli Flow

Symbolic synthesis of Clifford circuits and beyond

Formal Methods for Quantum Programs: A Survey

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