Clifford gates in the Holant framework

Cai, Jin-Yi; Guo, Heng; Williams, Tyson

doi:10.1016/j.tcs.2018.06.010

Cited by 6 publications

(6 citation statements)

References 35 publications

(55 reference statements)

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“…But 2uv +2u v becomes 2uv +2uv which is not yet trivial. This also compares to point 3 of Lemma 3.1 in [CGW18]. Part (b) follows individually for each term t plus t = π(t).…”

Section: Self-dual Forms and Probability Reduction To Rankmentioning

confidence: 79%

“…Such forms are called classical, reflecting the historical definition of a quadratic form as given by x Ax for some integer n × n matrix A that is symmetric-so that all cross terms have even coefficients. For Z 4 they coincide with those called affine in [CLX14,CGW18]. For contrast, we note the effect of using a controlled-phase gate:…”

Section: Quantum Stabilizer Circuitsmentioning

confidence: 87%

“…The connections used in our proof run through the real-time conversion of quantum circuits C to "phase polynomials" q C over Z K for K = 2 k , k ≥ 1 in [RC09,RCG18], which extended results by [DHH + 04] for k = 1, and the analysis of quadratic forms over Z 4 by Schmidt [Sch09] drawing on [Alb38,Bro72]. In the case of graph-state circuits and stabilizer circuits more generally, q C becomes a classical quadratic form over Z 4 , as treated also in [CGW18]. Our approach is related to ones involving Gauss sums [BvDR08, CCLL10, CGW18, Bk18] but exploits the availability of normal forms.…”

Section: Introductionmentioning

confidence: 83%

“…We seek even simpler and faster methods that lend themselves to further algorithmic properties, such as quick update when changes are made to C in the sense of "dynamic algorithms." We employ the theory of classical quadratic forms over Z 4 as developed by Schmidt [Sch09] and more recently by Cai, Guo, and Williams [CGW18]. The quadratic forms are built using the real-time algorithm of [RC09,RCG18] for computing what we call the additive partition polynomials q C for quantum circuits C that meet a mild "balance" condition.…”

Section: Quantum Stabilizer Circuitsmentioning

confidence: 99%

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Stabilizer Circuits, Quadratic Forms, and Computing Matrix Rank

Guan,

Regan

2019

Preprint

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We show that a form of strong simulation for n-qubit quantum stabilizer circuits C is computable in O(s + n ω ) time, where ω is the exponent of matrix multiplication. Solution counting for quadratic forms over F 2 is also placed into O(n ω ) time. This improves previous O(n 3 ) bounds. Our methods in fact show an O(n 2 )-time reduction from matrix rank over F 2 to computing p = | 0 n | C |0 n | 2 (hence also to solution counting) and a converse reduction that is O(s + n 2 ) except for matrix multiplications used to decide whether p > 0. The current best-known worst-case time for matrix rank is O(n ω ) over F 2 , indeed over any field, while ω is currently upper-bounded by 2.3728 . . . Our methods draw on properties of classical quadratic forms over Z 4 . We study possible distributions of Feynman paths in the circuits and prove that the differences in +1 vs. −1 counts and +i vs. −i counts are always 0 or a power of 2. Further properties of quantum graph states and connections to graph and matroid theory are discussed.

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“…But 2uv +2u v becomes 2uv +2uv which is not yet trivial. This also compares to point 3 of Lemma 3.1 in [CGW18]. Part (b) follows individually for each term t plus t = π(t).…”

Section: Self-dual Forms and Probability Reduction To Rankmentioning

confidence: 79%

Section: Quantum Stabilizer Circuitsmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 83%

Section: Quantum Stabilizer Circuitsmentioning

confidence: 99%

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Stabilizer Circuits, Quadratic Forms, and Computing Matrix Rank

Guan,

Regan

2019

Preprint

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“…For the reader familiar with quantum information theory, the affine functions correspond -up to scaling -to stabiliser states (cf. Section 4.2 and also independently [16]).…”

Section: Name #Csp(f)mentioning

confidence: 93%

A Full Dichotomy for $\hol^{c}$, Inspired by Quantum Computation

Backens¹

2021

SIAM J. Comput.

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The Complexity of Counting CSPd

Lin

2021

Theory Comput Syst

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Clifford gates in the Holant framework

Abstract: We show that the Clifford gates and stabilizer circuits in the quantum computing literature, which admit efficient classical simulation, are equivalent to affine signatures under a unitary condition. The latter is a known class of tractable functions under the Holant framework.

Cited by 6 publications

References 35 publications

Stabilizer Circuits, Quadratic Forms, and Computing Matrix Rank

Stabilizer Circuits, Quadratic Forms, and Computing Matrix Rank

A Full Dichotomy for $\hol^{c}$, Inspired by Quantum Computation

The Complexity of Counting CSPd

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