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Bounds on the power of constantdepth quantum circuits. Preprint: quantph/0312209
 In Proc. 15th International Symposium on on Fundamentals of Computation Theory (FCT 2005), volume 3623 of Lecture Notes in Computer Science
, 2004
"... We show that if a language is recognized within certain error bounds by constantdepth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, for 0 < ɛ ≤ δ ≤ 1, we define BQNC 0 ɛ,δ to be the class of languages recognized by constant dept ..."
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We show that if a language is recognized within certain error bounds by constantdepth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, for 0 < ɛ ≤ δ ≤ 1, we define BQNC 0 ɛ,δ to be the class of languages recognized by constant depth, polynomialsize quantum circuits with acceptance probability either < ɛ (for rejection) or ≥ δ (for acceptance). We show that BQNC 0 ɛ,δ ⊆ P, provided that 1 − δ ≤ 2 −2d (1 − ɛ), where d is the circuit depth. On the other hand, we adapt and extend ideas of Terhal & DiVincenzo [TD04] to show that, for any family F of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depthfive circuits over F is just as hard as computing these probabilities for arbitrary quantum circuits over F. In particular, this implies that NQNC 0 = NQACC = NQP = coC=P, where NQNC 0 is the constantdepth analog of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC ⊆ TC 0 [GHMP02]. 1
Quantum Lower Bounds for Fanout
, 2003
"... We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Tooli gates, and when they use only constantly many ancill. Under this constraint, t ..."
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Cited by 7 (2 self)
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We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Tooli gates, and when they use only constantly many ancill. Under this constraint, this bound is close to optimal. In the case of a nonconstant number a of ancill and n input qubits, we give a tradeo between a and the required depth, that results in a nontrivial lower bound for fanout when a = n 1 o(1) .
www.stacsconf.org DISTINGUISHING SHORT QUANTUM COMPUTATIONS
"... Abstract. Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of ..."
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Cited by 5 (0 self)
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Abstract. Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for QIP on constant depth circuits containing the unbounded fanout gate. These results are shown by reducing a QIPcomplete problem to a logarithmic depth version of itself using a parallelization technique. 1.
Implementing the fanout gate by a Hamiltonian
, 2003
"... We show that, for even n, evolving n qubits according to a simple Hamiltonian can be used to exactly implement an (n + 1)qubit parity gate, which is equivalent in constant depth to an (n + 1)qubit fanout gate. We also observe that evolving the Hamiltonian for three qubits results in an inversiono ..."
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We show that, for even n, evolving n qubits according to a simple Hamiltonian can be used to exactly implement an (n + 1)qubit parity gate, which is equivalent in constant depth to an (n + 1)qubit fanout gate. We also observe that evolving the Hamiltonian for three qubits results in an inversiononthreewayequality gate, which together with singlequbit operations is universal for quantum computation. 1
Quantum fanout is powerful
 Theory of Computing
"... We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, thr ..."
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We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fanin gates. If we allow arbitrary onequbit gates instead of a fixed basis, then these circuits can also be made exact in logstar depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth. 1
Universal Quantum Circuits
"... Abstract. We define and construct efficient depthuniversal and almostsizeuniversal quantum circuits. Such circuits can be viewed as generalpurpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth w ..."
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Abstract. We define and construct efficient depthuniversal and almostsizeuniversal quantum circuits. Such circuits can be viewed as generalpurpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blowup in the universal circuits constructed here. We prove that this construction is nearly optimal. 1
c ○ Rinton Press QUANTUM LOWER BOUNDS FOR FANOUT
, 2005
"... We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are pro ..."
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We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are proved. The main result is that parity (and hence fanout) requires log depth quantum circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancillæ. Under this constraint, this bound is close to optimal. In the case of a nonconstant number a of ancillæ and n input qubits, we give a tradeoff between a and the required depth, that results in a nonconstant lower bound for fanout when a = n 1−o(1). We also show that, regardless of the number of ancillæ arbitrary arity Toffoli gates cannot be simulated exactly by a constant depth circuit family with gates of bounded arity.
Computational Distinguishability of Quantum Channels
, 909
"... c ○ William Rosgen 2009I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. The computational problem of disti ..."
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c ○ William Rosgen 2009I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixedunitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixedunitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels. iii Acknowledgements I would like to thank my supervisor John Watrous for years of guidance, support, and insight. Without his help this would not have been possible. I would also like to thank the rest of my committee, Richard Cleve, Stephen Fenner, Achim Kempf, and Ben Reichardt, for providing helpful comments on an earlier draft of this thesis. I would also like to thank Lana for putting up with me during the writing of this thesis and supporting me throughout the process. v 4 The Close Images Problem 77 4.1 Logdepth mixedstate quantum circuits................... 78 4.2 QIP completeness of close images...................... 79