# Counting, fanout and the complexity of quantum ACC

@article{Green2002CountingFA, title={Counting, fanout and the complexity of quantum ACC}, author={Frederic Green and Steven Homer and Cristopher Moore and Chris Pollett}, journal={Quantum Inf. Comput.}, year={2002}, volume={2}, pages={35-65} }

We propose definitions of QAC0, the quantum analog of the classical class AC0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Modq gates are also allowed. We prove that parity or fanout allows us to construct quantum MODq gates in constant depth for any q, so QACC [2] = QACC. More generally, we show that for any q, p > 1, MODq is equivalent to MODp (up to constant depth and polynomial size). This implies that QAC0 with…

## 54 Citations

Quantum circuits with unbounded fan-out Robert Špalek

- Computer Science
- 2002

It is proposed that a new circuit class QNC 0 f — constant-depth quantum circuits with unbounded fan-out and a stronger version of QACC 0 is considered, which is related to an open problem of [6], however it is considered to be stronger than this.

A ug 2 00 2 Quantum circuits with unbounded fan-out

- Computer Science
- 2002

It is proposed that QTC 0 f = QACC 0, which was an open problem of [5], and it follows, that QNC 0 ⊆ Q NC 0 f ⊅ Q ACC 0, and an efficient method for performing commuting gates in parallel is described.

Quantum lower bounds for fanout

- Computer ScienceQuantum Inf. Comput.
- 2006

It is shown that, regardless of the number of ancillae arbitrary arity Toffoli gates cannot besimulated exactly by a constant depth circuit family with gates of bounded arity, and that parity requires log depth quantum circuits.

Bounds on the Power of Constant-Depth Quantum Circuits

- Computer ScienceFCT
- 2005

We show that if a language is recognized within certain error bounds by constant-depth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In…

Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates

- Computer ScienceQuantum Inf. Comput.
- 2013

This work shows that there exists a constant-depth quantum circuit with only two unbounded fan-out gates that is not strongly simulatable, unless P = PP, which is in contrast to the fact that any constant- depth quantum circuit without additional gates on an unbounded number of qubits is strongly and weakly simulatable.

Quantum Circuits with Unbounded Fan-out

- MathematicsSTACS
- 2003

It is demonstrated that the unbounded fan-out gate is very powerful and can approximate with polynomially small error the following gates: parity, mod, And, Or, majority, threshold, threshold[t], exact[q], and counting.

Bounds on the QAC0 Complexity of Approximating Parity

- Computer ScienceITCS
- 2021

The proofs use a new normal form for quantum circuits which may be of independent interest, and are based on reductions to the problem of constructing certain generalizations of the cat state which are named "nekomata" after an analogous cat y\=okai.

Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

- MathematicsSTOC
- 2019

The Parity Halving Problem is constructed by constructing a new problem in QNC^0, which is easier to work with, and it is proved that AC^0 lower bounds for this problem are proved, and that it reduces to the 2D HLF problem.

Quantum Fan-out is Powerful

- MathematicsTheory Comput.
- 2005

It is demonstrated that the unbounded fan-out gate is very powerful and can approximate with polynomially small error the follow- ing gates: parity, mod(q), And, Or, majority, threshold (t), exact(t), and Counting.

Perfect computational equivalence between quantum Turing machines and finitely generated uniform quantum circuit families

- MathematicsQuantum Inf. Process.
- 2009

The class of finitely generated uniform QCFs is perfectly equivalent to the class of polynomial-time QTMs and can exactly simulate each other; this naturally implies that BQP as well as ZQP and EQP equal the corresponding complexity classes of the finitely generate uniform Q CFs.

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