Power of Quantum Computation with Few Clean Qubits

Power of Quantum Computation with Few Clean Qubits∗ Keisuke Fujii1 , Hirotada Kobayashi2 , Tomoyuki Morimae3 , Harumichi Nishimura4 , Shuhei Tamate5 , and Seiichiro Tani6 1 2 3 4 5 6 The Hakubi Center for Advanced Research and Quantum Optics Group, Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto, Japan Principles of Informatics Research Division, National Institute of Informatics, Tokyo, Japan Advanced Scientific Research Leaders Development Unit, Gunma University, Kiryu, Gunma, Japan Department of Computer Science and Mathematical Informatics, Graduate School of Information Science, Nagoya University, Nagoya, Aichi, Japan Principles of Informatics Research Division, National Institute of Informatics, Tokyo, Japan NTT Communication Science Laboratories, NTT Corporation, Atsugi, Kanagawa, Japan Abstract This paper investigates the power of polynomial-time quantum computation in which only a very limited number of qubits are initially clean in the |0i state, and all the remaining qubits are initially in the totally mixed state. No initializations of qubits are allowed during the computation, nor are intermediate measurements. The main contribution of this paper is to develop unexpectedly strong error-reduction methods for such quantum computations that simultaneously reduce the number of necessary clean qubits. It is proved that any problem solvable by a polynomialtime quantum computation with one-sided bounded error that uses logarithmically many clean qubits is also solvable with exponentially small one-sided error using just two clean qubits, and with polynomially small one-sided error using just one clean qubit. It is further proved in the twosided-error case that any problem solvable by such a computation with a constant gap between completeness and soundness using logarithmically many clean qubits is also solvable with exponentially small two-sided error using just two clean qubits. If only one clean qubit is available, the problem is again still solvable with exponentially small error in one of the completeness and soundness and with polynomially small error in the other. An immediate consequence is that the Trace Estimation problem defined with fixed constant threshold parameters is complete for BQ[1] P and BQlog P, the classes of problems solvable by polynomial-time quantum computations with completeness 2/3 and soundness 1/3 using just one and logarithmically many clean qubits, respectively. The techniques used for proving the error-reduction results may be of independent interest in themselves, and one of the technical tools can also be used to show the hardness of weak classical simulations of one-clean-qubit computations (i.e., DQC1 computations). 1998 ACM Subject Classification F.1.2 Modes of Computation, F.1.3 Complexity Measures and Classes Keywords and phrases DQC1, quantum computing, complete problems, error reduction Digital Object Identifier 10.4230/LIPIcs.ICALP.2016.13 ∗ A full version [11] of this paper is available at arXiv.org e-Print archive, arXiv:1509.07276 [quant-ph]. EA TC S © Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani; licensed under Creative Commons License CC-BY 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Editors: Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi; Article No. 13; pp. 13:1–13:14 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 13:2 Power of Quantum Computation with Few Clean Qubits 1 Introduction 1.1 Background One of the most important goals in quantum information processing is to realize a quantum mechanical machine whose computational ability is superior to classical computers. The ultimate goal is, of course, to realize a large scale universal quantum computer, which still seems to be many years off despite extensive experimental efforts. Plenty of attention has thus been paid to “intermediate” (i.e., non-universal) models of quantum computation, which are somehow easier to physically implement. Such intermediate models do not offer universal quantum computation, but are believed to still be able to solve some problems that are hard for classical computers. The deterministic quantum computation with one quantum bit (DQC1 ), often mentioned as the one-clean-qubit model, is one of the most well-studied examples of such intermediate models. This model was introduced by Knill and Laflamme [15] to reflect some actual experimental setups such as nuclear magnetic resonance (NMR), where pure clean qubits are very hard to prepare and therefore are considered as very expensive resources. For example, in nuclear spin ensemble systems such as liquid state NMR systems, it is usually extremely hard, although not impossible, to polarize a spin (i.e., to initialize a qubit to state |0i), since energy scale of a nuclear spin qubit is quite small, while it is favorable for long coherence time. A DQC1 computation over w qubits starts with the initial state of
⊗(w−1) the totally mixed state except for a single clean qubit, namely, |0ih0| ⊗ I2 . After applying a polynomial-size unitary quantum circuit to this state, only a single output qubit is measured in the computational basis at the end of the computing in order to read out the computation result. No initializations of qubits are allowed during the computation, nor are intermediate measurements. The DQC1 model is believed not to have full computational power of the standard polynomial-time quantum computation, and is indeed strictly less powerful under some reasonable assumptions [5]. At first glance the model even looks easy to classically simulate and does not seem to offer any quantum advantage, partly because its highly-mixed initial state obviously lacks “quantumness” such as entanglement, coherence, and discord, which are widely believed to be origins of the power of quantum information processing, and also because any time-evolution over a single-qubit state or a totally mixed state is trivially simulatable by a classical computation. Nevertheless, the DQC1 model is not trivial, either, in the sense that it can efficiently solve several problems for which no efficient classical algorithms are known, such as estimating the spectral density [15], testing integrability [20], calculating the fidelity decay [19], approximating the Jones and HOMFLY polynomials [23, 13], and approximating an invariant of 3-manifolds [12]. As many of these problems have physically meaningful applications, the DQC1 model is one of the most important intermediate quantum computation models. Despite its importance explained thus far and the fact that tons of papers in physics have focused on it, very little has been studied on the genuinely complexity-theoretic aspects of the DQC1 model (to the best knowledge of the authors, no such studi (...truncated)