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arXiv:1605.00713 · Physical Review Letters

Efficient Quantum Pseudorandomness

Fernando G. S. L. Brandão, Aram W. Harrow, Michał Horodecki

Published Apr 29, 2016• 51 citations
arXivHTMLPDF

Abstract

Randomness is both a useful way to model natural systems and a useful tool for engineered systems, e.g., in computation, communication, and control. Fully random transformations require exponential time for either classical or quantum systems, but in many cases pseudorandom operations can emulate certain properties of truly random ones. Indeed, in the classical realm there is by now a well-developed theory regarding such pseudorandom operations. However, the construction of such objects turns out to be much harder in the quantum case. Here, we show that random quantum unitary time evolutions ("circuits") are a powerful source of quantum pseudorandomness. This gives for the first time a polynomial-time construction of quantum unitary designs, which can replace fully random operations in most applications, and shows that generic quantum dynamics cannot be distinguished from truly random processes. We discuss applications of our result to quantum information science, cryptography, and understanding the self-equilibration of closed quantum dynamics.

DOI
10.1103/physrevlett.116.170502

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Randomness is both a useful way to model natural systems and a useful tool for engineered systems, e.g., in computation, communication, and control. Fully random transformations require exponential time for either classical or quantum systems, but in many cases pseudorandom operations can emulate certain properties of truly random ones. Indeed, in the classical realm there is by now a well-developed theory regarding such pseudorandom operations. However, the construction of such objects turns out to be much harder in the quantum case. Here, we show that random quantum unitary time evolutions ("circuits") are a powerful source of quantum pseudorandomness. This gives for the first time a polynomial-time construction of quantum unitary designs, which can replace fully random operations in most applications, and shows that generic quantum dynamics cannot be distinguished from truly random processes. We discuss applications of our result to quantum information science, cryptography, and understanding the self-equilibration of closed quantum dynamics.
Quantum pseudorandomness has found applications in many areas of quantum information, ranging from entanglement theory, to models of scrambling phenomena in chaotic quantum systems, and, more recently, in the foundations of quantum cryptography. Kretschmer (TQC '21) showed that both pseudorandom states and pseudorandom unitaries exist even in a world without classical one-way functions. To this day, however, all known constructions require classical cryptographic building blocks which are themselves synonymous with the existence of one-way functions, and which are also challenging to realize on realistic quantum hardware. In this work, we seek to make progress on both of these fronts simultaneously -- by decoupling quantum pseudorandomness from classical cryptography altogether. We introduce a quantum hardness assumption called the Hamiltonian Phase State (HPS) problem, which is the task of decoding output states of a random instantaneous quantum polynomial-time (IQP) circuit. Hamiltonian phase states can be generated very efficiently using only Hadamard gates, single-qubit Z-rotations and CNOT circuits. We show that the hardness of our problem reduces to a worst-case version of the problem, and we provide evidence that our assumption is plausibly fully quantum; meaning, it cannot be used to construct one-way functions. We also show information-theoretic hardness when only few copies of HPS are available by proving an approximate ttt-design property of our ensemble. Finally, we show that our HPS assumption and its variants allow us to efficiently construct many pseudorandom quantum primitives, ranging from pseudorandom states, to quantum pseudoentanglement, to pseudorandom unitaries, and even primitives such as public-key encryption with quantum keys.
Can simple processes appear highly complex? Gowers (Comb. Prob. Comp. '96) conjectured that repeatedly composing local random reversible operations can yield global permutations that are indistinguishable from random. In this work, we study the unitary quantum analog of this question, in an attempt to make new progress on this longstanding conjecture. Our first result shows that statistical moment matching in the form of unitary designs does not generically lead to pseudorandomness---even for the simplest quantum processes: for every fixed ttt, we give an efficiently samplable family {νn}n\{ν_n\}_n{νn​}n​ of distributions on one- and two-qubit gates such that, after T=Ot(n2log⁡2n)T=O_t(n^2\log^2 n)T=Ot​(n2log2n) independent steps, the resulting nnn-qubit ensemble is an approximate unitary ttt-design with negligible error exp⁡(−Ω(log⁡2n))\exp(-Ω(\log^2 n))exp(−Ω(log2n)), yet an efficient quantum algorithm distinguishes it from random using only Ot(log⁡2n)O_t(\log^2 n)Ot​(log2n) queries. This refutes the unitary analog of the Hoory--Magen--Myers--Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.
Quantum pseudorandomness has found applications in many areas of quantum information, ranging from entanglement theory, to models of scrambling phenomena in chaotic quantum systems, and, more recently, in the foundations of quantum cryptography. Kretschmer (TQC '21) showed that both pseudorandom states and pseudorandom unitaries exist even in a world without classical one-way functions. To this day, however, all known constructions require classical cryptographic building blocks which are themselves synonymous with the existence of one-way functions, and which are also challenging to implement on realistic quantum hardware. In this work, we seek to make progress on both of these fronts simultaneously - by decoupling quantum pseudorandomness from classical cryptography altogether. We introduce a quantum hardness assumption called the Hamiltonian Phase State (HPS) problem, which is the task of decoding output states of a random instantaneous quantum polynomial-time (IQP) circuit. Hamiltonian phase states can be generated very efficiently using only Hadamard gates, single-qubit Z rotations and CNOT circuits. We show that the hardness of our problem reduces to a worst-case version of the problem, and we provide evidence that our assumption is plausibly fully quantum; meaning, it cannot be used to construct one-way functions. We also show information-theoretic hardness when only few copies of HPS are available by proving an approximate t-design property of our ensemble. Finally, we show that our HPS assumption and its variants allow us to efficiently construct many pseudorandom quantum primitives, ranging from pseudorandom states, to quantum pseudoentanglement, to pseudorandom unitaries, and even primitives such as public-key encryption with quantum keys.
This thesis is focused on the generation and understanding of particular kinds of quantum randomness. Randomness is useful for many tasks in physics and information processing, from randomized benchmarking , to black hole physics , as well demonstrating a so-called quantum speedup , and many other applications. On the one hand we explore how to generate a particular form of random evolution known as a t-design. On the other we show how this can also give instances for quantum speedup - where classical computers cannot simulate the randomness efficiently. We also show that this is still possible in noisy realistic settings. More specifically, this thesis is centered around three main topics. The first of these being the generation of epsilon-approximate unitary t-designs. In this direction, we first show that non-adaptive, fixed measurements on a graph state composed of poly(n,t,log(1/epsilon)) qubits, and with a regular structure (that of a brickwork state) effectively give rise to a random unitary ensemble which is a epsilon-approximate t-design. This work is presented in Chapter 3. Before this work, it was known that non-adaptive fixed XY measurements on a graph state give rise to unitary t-designs , however the graph states used there were of complicated structure and were therefore not natural candidates for measurement based quantum computing (MBQC), and the circuits to make them were complicated. The novelty in our work is showing that t-designs can be generated by fixed, non-adaptive measurements on graph states whose underlying graphs are regular 2D lattices. These graph states are universal resources for MBQC. Therefore, our result allows the natural integration of unitary t-designs, which provide a notion of quantum pseudorandomness which is very useful in quantum algorithms, into quantum algorithms running in MBQC. Moreover, in the circuit picture this construction for t-designs may be viewed as a constant depth quantum circuit, albeit with a polynomial number of ancillas. We then provide new constructions of epsilon-approximate unitary t-designs both in the circuit model and in MBQC which are based on a relaxation of technical requirements in previous constructions. These constructions are found in Chapters 4 and 5.
The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a 222-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary kkk-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions.