Computational Distinguishability of Quantum Channels by William Rosgen A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada, 2009 c William Rosgen 2009 I 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. ii Abstract The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the well-known 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 mixed-unitary 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 mixed-unitary 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 Contents List of Figures xii List of Symbols xiii 1 Introduction 1 1.1 Overview . .2 1.2 Quantum information . .6 1.3 Classes of quantum channels . 18 2 Quantum Computational Complexity 25 2.1 Quantum circuits . 26 2.2 Quantum complexity classes . 35 3 Measures for Quantum Information 41 3.1 Entropy . 42 3.2 Schatten p-norms . 45 3.3 The classical capacity of a quantum channel . 47 3.4 The trace norm . 53 3.5 The diamond norm . 57 3.6 Fidelity . 64 3.7 Polarization of the diamond norm . 69 3.8 Conclusion . 76 vii 4 The Close Images Problem 77 4.1 Log-depth mixed-state quantum circuits . 78 4.2 QIP completeness of close images . 79 4.3 The swap test . 83 4.4 Reduction to logarithmic depth . 87 4.5 Correctness of the reduction . 93 4.6 Conclusion . 98 5 Distinguishability of Quantum Computations 99 5.1 Overview of distinguishability problems . 100 5.2 Quantum circuit distinguishability . 102 5.3 QIP protocol . 104 5.4 Reduction from Close Images . 107 5.5 Correctness of the reduction . 110 5.6 Distinguishing log-depth computations . 115 5.7 Conclusion . 116 6 Degradable and Antidegradable Channels 117 6.1 Degradable and antidegradable channels . 118 6.2 Simulation by a degradable channel . 119 6.3 Distinguishing degradable channels . 121 6.4 Simulation by an antidegradable channel . 123 6.5 Distinguishing antidegradable channels . 126 6.6 Conclusion . 127 7 Mixed-Unitary Channels 129 7.1 Mixed-unitary channels . 130 7.2 Unital channels . 132 7.3 Mixed-unitary approximation . 134 7.4 Properties of the constructed channel . 142 viii 7.5 Multiplicativity of mixed-unitary transformations . 145 7.6 Mixed-unitaries and minimum output entropy . 147 7.7 Circuit constructions . 149 7.8 QIP-completeness of distinguishing mixed-unitary circuits . 156 7.9 Conclusion . 160 8 Conclusion 161 Bibliography 163 Index 177 ix List of Figures 1.1 Optimal strategy for channel distinguishability . .3 1.2 Reductions presented in the thesis . .4 2.1 An example quantum circuit . 27 2.2 Gates in the unitary circuit model . 29 2.3 Simulation of the swap gate with three controlled-not gates . 29 2.4 Controlled-U gate . 30 2.5 Non-unitary gates in the mixed state circuit model . 31 2.6 Simulations of three of the gates in the mixed-state circuit model . 31 2.7 Simulating a channel with a unitary circuit . 32 2.8 Log-depth implementation of controlled operation on n qubits . 34 2.9 Constant depth implementation of controlled operation on n qubits . 35 2.10 Known relationships between complexity classes . 36 2.11 A three message quantum interactive proof system . 38 3.1 Circuits output by the construction in Lemma 3.26 . 72 3.2 Circuits output by the construction in Lemma 3.28 . 74 4.1 Transformations in a quantum interactive proof system . 80 4.2 Reduction from QIP to Close Images ..................... 81 4.3 Constant-depth implementation of a swap gate . 84 4.4 Circuit implementation of the swap test . 84 4.5 Decomposition of a circuit into constant depth pieces . 88 xi 4.6 Testing procedure used in reduction to log-depth circuits . 89 4.7 Overview of the output spaces of the constructed log-depth circuits . 90 4.8 The output of the reduction to log-depth circuits . 91 5.1 Complexity classes and distinguishability problems . 101 5.2 Input circuits for the reduction . 108 5.3 Circuit to apply either input circuit based on a control qubit . 108 5.4 Circuits output by the reduction . 109 6.1 Reduction to a degradable channel . 120 6.2 Degrading channel for the channel in Figure 6.1 . 121 6.3 Reduction to an antidegradable channel . 124 6.4 Anti-degrading channel for the channel in Figure 6.3 . 125 7.1 Channel to be approximated by a mixed-unitary . 135 7.2 One stage of the circuit for the ancilla simulation procedure . 153 7.3 Circuit performing the ancilla simulation procedure . 154 7.4 The mixed-unitary circuit that simulates the original circuit . 154 xii List of Symbols 11˜ H completely mixed state on H, 11˜ H = 11H= dim H δij Kronecker delta function: δij = 1 if i = j and 0 otherwise F(ρ, σ) fidelity of states ρ and σ Fmax(Φ, Ψ) Maximum output fidelity of channels Φ and Ψ H, K,... finite dimensional Hilbert spaces isomorphism between Hilbert spaces log base-two logarithm k·k diamond norm k·kp Schatten p-norm k·ktr trace norm ν(·) Maximum output p-norm S(ρ) von Neumann entropy of the state ρ Smin(Φ) minimum output entropy of the channel Φ Classes of Operators D(H) density operators on H L(H, K) set of all linear operators from H to K T(H, K) set of all channels from L(H) to L(K) U(H) unitary operators on H U(H, K) isometries mapping H to K xiii Complexity Classes BQP quantum efficiently solvable problems EXP classicaly solvable problems in exponential time NP classicaly efficiently verifiable problems P classicaly efficiently solvable problems PSPACE classicaly solvable problems in polynomial space QIP quantum efficiently interactively verifiable problems QMA quantum efficiently verifiable problems Operators 11H identity operator on L(H, H) H Hadamard matrix IH identity transformation on L(H) trK Partial trace over the system in K W Swap operation: Wjaijbi = jbijai X Pauli X matrix Y Pauli Y matrix Z Pauli Z matrix Quantum Circuits j0i Introduction of ancillary qubits in j0i state Controlled-not gate t d t Controlled U gate U D Completely dephasing channel D(jiihjj) = δijjiihjj xiv H Hadamard gate 7 Measurement in the computational basis fj0i, j1ig N Completely depolarizing channel N(ρ) = 11=d Partial trace ? Two-qubit swap gate W Multi-qubit swap gate W(jaijbi) = jbijai X Pauli X gate Z Pauli Z gate Computational Problems CI Close Images Problem QCD Quantum Circuit Distinguishability problem xv Chapter 1 Introduction Distinguishing two quantum channels is one of the most important tasks in quantum information. This is the problem of determining if there is an input state on which the two channels to produce output states that are distinguishable. When this is phrased as a computational problem it is complete for the complexity class QIP of problems that have quantum interactive proof systems. This problem seems to be computationally much more difficult than other variants of the problem, such as distinguishing classical circuits or distinguishing unitary quantum circuits. In light of this hardness, it is natural to consider restricted versions of the problem. Many of these special cases are also hard: reductions can be found to some of the more interesting classes of quantum channels. These results suggest that this problem is not likely to be tractable even on many of the restricted channels that can be realized by experiment. This is, however, not a surprise: distinguishing two channels is a restricted version of quantum process tomography, which is computationally intractable for large systems. These reductions provide simulations of general quantum channels by channels in restricted classes. While these simulations do not accurately model all aspects of the original channel, the constructed simulations do share many properties with the original channel. Many of these results can be applied outside the narrow focus of distinguishing quantum channels: it is hoped that these techniques will prove useful for a number of problems in quantum information theory. Contents 1.1 Overview . .2 1.2 Quantum information . .6 1.2.1 Hilbert spaces . .7 1 1.2.2 Pure states . .9 1.2.3 Linear operators . 10 1.2.4 Mixed states .