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… 

Figures from this paper

Quantum circuits with unbounded fan-out Robert Špalek
  • Computer Science
  • 2002
TLDR
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
TLDR
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
TLDR
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
...
...

References

SHOWING 1-10 OF 46 REFERENCES
Quantum Circuits: Fanout, Parity, and Counting
  • C. Moore
  • Physics
    Electron. Colloquium Comput. Complex.
  • 1999
TLDR
It is shown that it is possible to make a `cat' state on n qubits in constant depth if and only if the authors can construct a parity or Mod-2 gate in constantDepth; therefore, any circuit class that can fan out a qubit to n copies in constant Depth also includes QACC^0[2].
On the complexity of quantum ACC
TLDR
The result resolves the question, proving that QAC/sub wf//sup (0)/=QACC[q] QACC for all q, and develops techniques for proving upper bounds for QACC in terms of related language classes.
Quantum Computability
TLDR
It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in P and PSPACE.
Quantum NP is Hard for PH
TLDR
It is proved that in fact NQP and coC=P are the same class, by showing that determining whether a quantum computation has a non-zero probability of accepting is hard for coC-P.
Elementary gates for quantum computation.
TLDR
U(2) gates are derived, which derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number of unitary operations on arbitrarily many bits.
Parallel Quantum Computation and Quantum Codes
TLDR
While it is noted the exact quantum Fourier transform can be parallelized to linear depth, it is conjecture that neither it nor a simpler "staircase" circuit can be Parallelized to less than this.
Circuits with monoidal gates
The problem of evaluating a circuit whose wires carry values from a fixed finite monoid M and whose non-input gates perform the monoid's operation is a natural extension to the well studied word
Quantum Circuit Complexity
  • A. Yao
  • Computer Science
    FOCS
  • 1993
TLDR
It is shown that any function computable in polynomial time by a quantum Turing machine has aPolynomial-size quantum circuit, and this result enables us to construct a universal quantum computer which can simulate a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them.
Complexity limitations on quantum computation
  • L. Fortnow, J. Rogers
  • Computer Science
    Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247)
  • 1998
We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum
NQPC = co-C=P
...
...