Abstract
The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high-energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many-body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a shortcut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary k-designs, ensembles that capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind. Moreover, our results apply under a strong definition of quantum complexity based on optimal distinguishing measurements.Received 13 January 2021Accepted 21 May 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030316Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasComputational complexityQuantum correlations in quantum informationQuantum entanglementQuantum measurementsPhysical SystemsQuantum chaotic systemsStrongly correlated systemsTechniquesMany-body techniquesRandom matrix theoryQuantum InformationParticles & FieldsStatistical PhysicsCondensed Matter, Materials & Applied Physics
Highlights
We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind
We rigorously establish a growth of the quantum complexity in the time evolution of a number of models
We prove that with overwhelming probability, an element sampled from an approximate unitary k-design has a strong complexity that scales at least linearly in k
Summary
The complexity of a computation is a measure of the resources needed to perform the computation. While Corollary 4 does not by itself strongly constrain how these high-complexity unitary transformations are distributed geometrically within the n-qudit unitary group, we are able to prove a stronger result: An approximate k-design contains exponentially many (in k) high-complexity unitaries whose pairwise distance (i.e., the distance between any pair of unitaries) is almost maximal in the diamond norm This stronger statement rules out the possibility that most of the high-complexity unitaries reside inside a few tightly packed clusters within U(d). [27] shows that, for a particular class of stochastic Hamiltonians, evolution time linear in k sou(√ffince).s to generate Corollary 4 approximate k-designs for implies that with k= high probability the complexity grows linearly in time, at least for a while.
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