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"Quantum Computational Complexity"
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Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically, we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, either exactly in total variation distance or even approximately up to 41 per cent multiplicative error in the probabilities, then the infinite tower of classical complexity classes known as the polynomial hierarchy would collapse to its third level. We mention some further results on the classical simulation properties of IQP circuit families, in particular showing that if the output distribution results from measurements on only O(log n) lines then it may, in fact, be classically efficiently sampled.
Journal Article
Matchgates and classical simulation of quantum circuits
Let G(A, B) denote the two-qubit gate that acts as the one-qubit SU(2) gates A and B in the even and odd parity subspaces, respectively, of two qubits. Using a Clifford algebra formalism, we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allow the same gates to act only on n.n. and next n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan-Wigner representation of a Clifford algebra, we show how one may generalize the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.
Journal Article
A new look at the C 0-formulation of the strong cosmic censorship conjecture
We examine the
C
0
-formulation of the strong cosmic censorship conjecture (SCC) from a quantum complexity-theoretic perspective and argue that for generic black hole parameters as initial conditions for the Einstein equations, corresponding to the expected geometry of a hyperbolic black hole, the metric is
C
0
-extendable to a larger Lorentzian manifold across the Cauchy horizon. To demonstrate the pathologies associated with a hypothetical validity of the
C
0
SCC, we prove it violates the ‘complexity = volume’ conjecture for a low-temperature hyperbolic AdS
d
+1
black hole dual to a CFT living on a (
d
− 1)-dimensional hyperboloid
H
d
−1
, where in order to preserve the gauge/gravity duality we impose a lower bound on the interior metric extendability of order the classical recurrence time.
Journal Article
Temporally unstructured quantum computation
We examine theoretic architectures and an abstract model for a restricted class of quantum computation, called here temporally unstructured ('instantaneous') quantum computation because it allows for essentially no temporal structure within the quantum dynamics. Using the theory of binary matroids, we argue that the paradigm is rich enough to enable sampling from probability distributions that cannot, classically, be sampled efficiently and accurately. This paradigm also admits simple interactive proof games that may convince a sceptic of the existence of truly quantum effects. Furthermore, these effects can be created using significantly fewer qubits than are required for running Shor's algorithm.
Journal Article
Matchgate and space-bounded quantum computations are equivalent
Matchgates are an especially multiflorous class of two-qubit nearest-neighbour quantum gates, defined by a set of algebraic constraints. They occur for example in the theory of perfect matchings of graphs, non-interacting fermions and one-dimensional spin chains. We show that the computational power of circuits of matchgates is equivalent to that of space-bounded quantum computation with unitary gates, with space restricted to being logarithmic in the width of the matchgate circuit. In particular, for the conventional setting of polynomial-sized (logarithmic-space generated) families of matchgate circuits, known to be classically simulatable, we characterize their power as coinciding with polynomial-time and logarithmic-space-bounded universal unitary quantum computation.
Journal Article
Introduction to Quantum Algorithms for Physics and Chemistry
This chapter introduces the basic concepts of digital quantum simulation. The study of the computational complexity of problems in quantum simulation helps us better understand how quantum computers can surpass classical computers. The chapter briefly summarizes a few important examples of complexity classes of decision problems. Quantum algorithms are procedures for applying elementary quantum logic gates to complete certain unitary transformations of the input state. The steps involved in carrying out a digital quantum simulation consist of three parts: state preparation, time evolution, and measurement of observables. The chapter provides an overview of state preparation and simulation of time evolution. The use of Suzuki–Trotter formulas in quantum simulation for time‐independent sparse Hamiltonians is reviewed. The chapter reviews a method to effect nondestructive measurements of constants of the motion within the adiabatic model.
Book Chapter
On a class of quantum Turing machine halting deterministically
We consider a subclass of quantum Turing machines （QTM）, named stationary rotational quantum Turing machine （SR-QTM）, which halts deterministically and has deterministic tape head position. A quantum state transition diagram （QSTD） is proposed to describe SR-QTM. With QSTD, we construct a SR-QTM which is universal for all near-trivial transformations. This indicates there exists a QTM which is universal for the above subclass. Finally we show that SR-QTM is computational equivalent with ordinary QTM in the bounded error setting. It can be seen that SR-QTMs have deterministic tape head position and halt deterministically, and thus the halting scheme problem will not exist for this class of QTMs.
Journal Article
Computing quantum discord is NP-complete
We study the computational complexity of quantum discord (a measure of quantum correlation beyond entanglement), and prove that computing quantum discord is NP-complete. Therefore, quantum discord is computationally intractable: the running time of any algorithm for computing quantum discord is believed to grow exponentially with the dimension of the Hilbert space so that computing quantum discord in a quantum system of moderate size is not possible in practice. As by-products, some entanglement measures (namely entanglement cost, entanglement of formation, relative entropy of entanglement, squashed entanglement, classical squashed entanglement, conditional entanglement of mutual information, and broadcast regularization of mutual information) and constrained Holevo capacity are NP-hard/NP-complete to compute. These complexity-theoretic results are directly applicable in common randomness distillation, quantum state merging, entanglement distillation, superdense coding, and quantum teleportation; they may offer significant insights into quantum information processing. Moreover, we prove the NP-completeness of two typical problems: linear optimization over classical states and detecting classical states in a convex set, providing evidence that working with classical states is generically computationally intractable.
Journal Article
Statistical Approach to Quantum Field Theory
This book opens with a self-contained introduction to path integrals in Euclidean quantum mechanics and statistical mechanics, and moves on to cover lattice field theory, spin systems, gauge theories and more. Each chapter ends with illustrative problems.
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