Techniques to Reduce $π/4$-Parity-Phase Circuits, Motivated by the ZX Calculus

de Beaudrap, Niel; Bian, Xiaoning; Wang, Quanlong

doi:10.48550/arxiv.1911.09039

Cited by 8 publications

(11 citation statements)

References 13 publications

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“…Like the other exact synthesis algorithms, the running time of this parallel algorithm is exponential in the number of qubits as well as depth of the circuit. The problem of reducing T-count with the help of ZX calculus has also been studied [dBBW19] and in some cases it gives better results than previously known ones.…”

Section: Related Workmentioning

confidence: 99%

A polynomial time and space heuristic algorithm for T-count

Mosca,

Mukhopadhyay

2020

Preprint

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An important part of reaping computational advantage from a quantum computer is to reduce the quantum resources needed to implement a desired quantum algorithm. Quantum algorithms that are too large to be practical on noisy intermediate scale quantum (NISQ) devices will require fault-tolerant error correction. This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set.More specifically, in this paper we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford+T gate set. The Clifford+T gate set is a universal gate set and in this group, using state-of-the-art surface codes, the T gate is by far the most expensive component to implement fault-tolerantly. So it is important to minimize the number of T gates necessary for a fault-tolerant implementation. Our primary interest is to compute a circuit for a given n-qubit unitary U , using the minimum possible number of T gates (called the T-count of U ). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of U . In its decision version the goal is to decide if the T-count is at most some positive integer m. Given an oracle for COUNT-T, we can compute a T-optimal circuit in time polynomial in the T-count and dimension of U . We give a provable classical algorithm that solves COUNT-T (decision) in time O N 2(c−1)⌈ m c ⌉ poly(m, N ) and space O N 2⌈ m c ⌉ poly(m, N ) , where N = 2 n and c ≥ 2. We also introduce an asymptotically faster multiplication method that shaves a factor of N 0.7457 off of the overall complexity.Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that solves COUNT-T (optimization) with both space and time poly(m, N ). While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent) , there is a large improvement by going from exponential to polynomial scaling with m. We implemented our heuristic algorithm with up to 4 qubit unitaries and obtained a significant improvement in time as well as T-count.

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Section: Related Workmentioning

confidence: 99%

A polynomial time and space heuristic algorithm for T-count

Mosca,

Mukhopadhyay

2020

Preprint

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“…Subject to the parameterised model of Eqns. (23), one may show that Eqn. (22b) necessarily holds if the green and red nodes form special commutative dagger-Frobenius algebras (this is an easy corollary of Lemma C.8, on page 21).…”

Section: Suppose That We Consider a Parameterised Model [[ • ]mentioning

confidence: 98%

“…This derivation features a phase gadget [20,23], a ZX term of independent interest which denotes a diagonal operation inducing a relative phase e iθ on standard basis states |x for which x • z ≡ 1 (mod 2) for some z ∈ {0, 1} n . One may show that the gadgets are precisely unitary in [[ • ]] ν by induction on n 1:…”

Section: Features Of the Well-tempered Calculimentioning

confidence: 99%

“…Specifically, the objects represented by ZX diagrams are in general tensor networks, in which any arrow of time is merely imposed by convention. As a result of this flexibility, it has found to be productive in application to quantum technologies, for error correction [16][17][18] and circuit optimisation problems such as reduction of phase gates and CNOT allocation [19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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Well-tempered ZX and ZH Calculi

de Beaudrap

2020

Preprint

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The ZX calculus is a mathematical tool to represent and analyse quantum operations by manipulating diagrams which in effect represent tensor networks. Two families of nodes of these networks are ones which commute with either Z rotations or X rotations, usually called "green nodes" and "red nodes" respectively. The original formulation of the ZX calculus was motivated in part by properties of the algebras formed by the green and red nodes: notably, that they form a bialgebra -but only up to scalar factors. As a consequence, the diagram transformations and notation for certain unitary operations involve "scalar gadgets" which denote contributions to a normalising factor. We present renormalised generators for the ZX calculus, which form a bialgebra precisely. As a result, no scalar gadgets are required to represent the most common unitary transformations, and the corresponding diagram transformations are generally simpler. We also present a similar renormalised version of the ZH calculus. We obtain these results by an analysis of conditions under which various "idealised" rewrites are sound, leveraging the existing presentations of the ZX and ZH calculi.

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“…The name 'phase gadget' refers to a particular configuration of spiders in the ZXcalculus, which, in our setting, corresponds to spiders measured in the YZ plane. Phase gadgets have been used particularly in the study of circuit optimisation [12,19,35].…”

Section: Phase-gadget Formmentioning

confidence: 99%

There and back again: A circuit extraction tale

Backens,

Miller-Bakewell,

de Felice

et al. 2020

Preprint

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Translations between the quantum circuit model and the measurementbased one-way model are useful for verification and optimisation of quantum computations. They make crucial use of a property known as gflow. While gflow is defined for one-way computations allowing measurements in three different planes of the Bloch sphere, most research so far has focused on computations containing only measurements in the XY-plane. Here, we give the first circuit-extraction algorithm to work for one-way computations containing measurements in all three planes and having gflow. The algorithm is efficient and the resulting circuits do not contain ancillae. One-way computations are represented using the ZX-calculus, hence the algorithm also represents the most general known procedure for extracting circuits from ZX-diagrams. In developing this algorithm, we generalise several concepts and results previously known for computations containing only XY-plane measurements. We bring together several known rewrite rules for measurement patterns and formalise them in a unified notation using the ZX-calculus. These rules are used to simplify measurement patterns by reducing the number of qubits while preserving both the semantics and the existence of gflow. The results can be applied to circuit optimisation by translating circuits to patterns and back again.

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Techniques to Reduce $π/4$-Parity-Phase Circuits, Motivated by the ZX Calculus

Cited by 8 publications

References 13 publications

A polynomial time and space heuristic algorithm for T-count

A polynomial time and space heuristic algorithm for T-count

Well-tempered ZX and ZH Calculi

There and back again: A circuit extraction tale

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