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arXiv:2206.15122 · Leibniz international proceedings in informatics

Space-Bounded Unitary Quantum Computation with Postselection

Seiichiro Tani

Published Jan 1, 2022• 0 citations
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Abstract

Space-bounded computation has been a central topic in classical and quantum complexity theory. In the quantum case, every elementary gate must be unitary. This restriction makes it unclear whether the power of space-bounded computation changes by allowing intermediate measurement. In the bounded error case, Fefferman and Remscrim [STOC 2021, pp.1343--1356] and Girish, Raz and Zhan~[ICALP 2021, pp.73:1--73:20] recently provided the break-through results that the power does not change. This paper shows that a similar result holds for space-bounded quantum computation with postselection. Namely, it is proved possible to eliminate intermediate postselections and measurements in the space-bounded quantum computation in the bounded-error setting. Our result strengthens the recent result by Le Gall, Nishimura and Yakaryilmaz~[TQC 2021, pp.10:1--10:17] that logarithmic-space bounded-error quantum computation with intermediate postselections and measurements is equivalent in computational power to logarithmic-space unbounded-error probabilistic computation. As an application, it is shown that bounded-error space-bounded one-clean qubit computation (DQC1) with postselection is equivalent in computational power to unbounded-error space-bounded probabilistic computation, and the computational supremacy of the bounded-error space-bounded DQC1 is interpreted in complexity-theoretic terms.

DOI
10.4230/lipics.mfcs.2022.81

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Space-bounded computation has been a central topic in classical and quantum complexity theory. In the quantum case, every elementary gate must be unitary. This restriction makes it unclear whether the power of space-bounded computation changes by allowing intermediate measurement. In the bounded error case, Fefferman and Remscrim [STOC 2021, pp.1343--1356] and Girish, Raz and Zhan~[ICALP 2021, pp.73:1--73:20] recently provided the break-through results that the power does not change. This paper shows that a similar result holds for space-bounded quantum computation with postselection. Namely, it is proved possible to eliminate intermediate postselections and measurements in the space-bounded quantum computation in the bounded-error setting. Our result strengthens the recent result by Le Gall, Nishimura and Yakaryilmaz~[TQC 2021, pp.10:1--10:17] that logarithmic-space bounded-error quantum computation with intermediate postselections and measurements is equivalent in computational power to logarithmic-space unbounded-error probabilistic computation. As an application, it is shown that bounded-error space-bounded one-clean qubit computation (DQC1) with postselection is equivalent in computational power to unbounded-error space-bounded probabilistic computation, and the computational supremacy of the bounded-error space-bounded DQC1 is interpreted in complexity-theoretic terms.
General quantum computation consists of unitary operations and also measurements. It is well known that intermediate quantum measurements can be deferred to the end of the computation, resulting in an equivalent purely unitary computation. While time efficient, this transformation blows up the space to linear in the running time, which could be super-polynomial for low-space algorithms. Fefferman and Remscrim (STOC'21) and Girish, Raz and Zhan (ICALP'21) show different transformations which are space efficient, but blow up the running time by a factor that is exponential in the space. This leaves the case of algorithms with small-but-super-logarithmic space as incurring a large blowup in either time or space complexity. We show that such a blowup is likely inherent, demonstrating that any "black-box" transformation which removes intermediate measurements must significantly blow up either space or time.
This paper investigates the power of polynomial-time quantum computation in which only a very limited number of qubits are initially clean in the |0> state, and all the remaining qubits are initially in the totally mixed state. No initializations of qubits are allowed during the computation, nor intermediate measurements. The main results of this paper are unexpectedly strong error-reducible properties of such quantum computations. It is proved that any problem solvable by a polynomial-time quantum computation with one-sided bounded error that uses logarithmically many clean qubits can also be 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 case of two-sided bounded error that any problem solvable by such a computation with a constant gap between completeness and soundness using logarithmically many clean qubits can also be 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 polynomially small error in the other. As an immediate consequence of the above result for the two-sided-error case, it follows that the TRACE ESTIMATION problem defined with fixed constant threshold parameters is complete for the classes of problems solvable by polynomial-time quantum computations with completeness 2/3 and soundness 1/3 using logarithmically many clean qubits and just one clean qubit. 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).
A foundational result in the theory of quantum computation, known as the "principle of safe storage," shows that it is always possible to take a quantum circuit and produce an equivalent circuit that makes all measurements at the end of the computation. While this procedure is time efficient, meaning that it does not introduce a large overhead in the number of gates, it uses extra ancillary qubits, and so is not generally space efficient. It is quite natural to ask whether it is possible to eliminate intermediate measurements without increasing the number of ancillary qubits. We give an affirmative answer to this question by exhibiting a procedure to eliminate all intermediate measurements that is simultaneously space efficient and time efficient. In particular, this shows that the definition of a space-bounded quantum complexity class is robust to allowing or forbidding intermediate measurements. A key component of our approach, which may be of independent interest, involves showing that the well-conditioned versions of many standard linear-algebraic problems may be solved by a quantum computer in less space than seems possible by a classical computer.
Intermediate measurements let quantum computations discard information and reuse their workspace. Deferring all measurements can require storing their entire history, which need not preserve logarithmic space. Fefferman and Remscrim proved that measurements can nevertheless be eliminated with two-sided bounded error, and asked whether the same holds with one-sided error [FR21]. We prove RQLΓ = RQULΓ for the standard gate set Γ = {H, T, CNOT}, preserving polynomial time, logarithmic space, and exactly zero acceptance on no-instances. Our proof extends the density-matrix doubling method of Girish, Raz and Zhan [ GRZ21 ] to general channels, including resets and classical memory. We also prove measurement elimination for broader finite gate sets with exact inverses, including suitable gates with transcendental entries, and establish gate-set independence for a specified family of algebraic gate sets. A separate history-checking construction proves QMAL1,G = QUMAL1,G under explicit gate assumptions: measurements can also be eliminated from logarithmic-space quantum verification while preserving perfect completeness
Motivated by understanding the power of quantum computation with restricted number of qubits, we give two complete characterizations of unitary quantum space bounded computation. First we show that approximating an element of the inverse of a well-conditioned efficiently encoded 2k(n)×2k(n)2^{k(n)}\times 2^{k(n)}2k(n)×2k(n) matrix is complete for the class of problems solvable by quantum circuits acting on O(k(n))\mathcal{O}(k(n))O(k(n)) qubits with all measurements at the end of the computation. Similarly, estimating the minimum eigenvalue of an efficiently encoded Hermitian 2k(n)×2k(n)2^{k(n)}\times 2^{k(n)}2k(n)×2k(n) matrix is also complete for this class. In the logspace case, our results improve on previous results of Ta-Shma [STOC '13] by giving new space-efficient quantum algorithms that avoid intermediate measurements, as well as showing matching hardness results. Additionally, as a consequence we show that PreciseQMA, the version of QMA with exponentially small completeness-soundess gap, is equal to PSPACE. Thus, the problem of estimating the minimum eigenvalue of a local Hamiltonian to inverse exponential precision is PSPACE-complete, which we show holds even in the frustration-free case. Finally, we can use this characterization to give a provable setting in which the ability to prepare the ground state of a local Hamiltonian is more powerful than the ability to prepare PEPS states. Interestingly, by suitably changing the parameterization of either of these problems we can completely characterize the power of quantum computation with simultaneously bounded time and space.
Post-selection, the power of discarding all runs of a computation in which an undesirable event occurs, is an influential concept introduced to the field of quantum complexity theory by Aaronson (Proceedings of the Royal Society A, 2005). In the present paper, we initiate the study of post-selection for space-bounded quantum complexity classes. Our main result shows the identity PostBQL=PL\sf PostBQL=PLPostBQL=PL, i.e., the class of problems that can be solved by a bounded-error (polynomial-time) logarithmic-space quantum algorithm with post-selection (PostBQL\sf PostBQLPostBQL) is equal to the class of problems that can be solved by unbounded-error logarithmic-space classical algorithms (PL\sf PLPL). This result gives a space-bounded version of the well-known result PostBQP=PP\sf PostBQP=PPPostBQP=PP proved by Aaronson for polynomial-time quantum computation. As a by-product, we also show that PL\sf PLPL coincides with the class of problems that can be solved by bounded-error logarithmic-space quantum algorithms that have no time bound.