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JHEP06(2016)001

Published for SISSA by Springer

Received: April 1, 2016

Revised: May 18, 2016

Accepted: May 19, 2016

Published: June 1, 2016

Dual ﬁeld theories of quantum computation

Vitaly Vanchurin

Department of Physics and Astronomy, University of Minnesota,

1023 University Dr., Duluth, MN 55812, U.S.A.

E-mail: vvanchur@d.umn.edu

Abstract: Given two quantum states of N q-bits we are interested to ﬁnd the shortest

quantum circuit consisting of only one- and two- q-bit gates that would transfer one state

into another. We call it the quantum maze problem for the reasons described in the paper.

We argue that in a large N limit the quantum maze problem is equivalent to the problem

of ﬁnding a semiclassical trajectory of some lattice ﬁeld theory (the dual theory) on an

N + 1 dimensional space-time with geometrically ﬂat, but topologically compact spatial

slices. The spatial fundamental domain is an N dimensional hyper-rhombohedron, and the

temporal direction describes transitions from an arbitrary initial state to an arbitrary target

state and so the initial and ﬁnal dual ﬁeld theory conditions are described by these two

quantum computational states. We ﬁrst consider a complex Klein-Gordon ﬁeld theory and

argue that it can only be used to study the shortest quantum circuits which do not involve

generators composed of tensor products of multiple Pauli Z matrices. Since such situation

is not generic we call it the Z-problem. On the dual ﬁeld theory side the Z-problem

corresponds to massless excitations of the phase (Goldstone modes) that we attempt to ﬁx

using Higgs mechanism. The simplest dual theory which does not suﬀer from the massless

excitation (or from the Z-problem) is the Abelian-Higgs model which we argue can be

used for ﬁnding the shortest quantum circuits. Since every trajectory of the ﬁeld theory

is mapped directly to a quantum circuit, the shortest quantum circuits are identiﬁed with

semiclassical trajectories. We also discuss the complexity of an actual algorithm that uses

a dual theory prospective for solving the quantum maze problem and compare it with

a geometric approach. We argue that it might be possible to solve the problem in sub-

exponential time in 2

, but for that we must consider the Klein-Gordon theory on curved

spatial geometry and/or more complicated (than N-torus) topology.

Keywords: Field Theories in Higher Dimensions, Lattice Quantum Field Theory

ArXiv ePrint: 1603.07982

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP06(2016)001

JHEP06(2016)001

Contents

1 Introduction 1

2 Path integrals for dual theories 2

3 Klein-Gordon dual theory 4

3.1 Dual lattice ﬁeld theory 5

3.2 Classical ﬁeld theory solutions 8

4 Quantum circuits from classical solutions 9

5 Abelian-Higgs dual theory 12

5.1 The Z-problem or massless excitations 12

5.2 Non-relativistic limit of ﬁeld theories 14

6 Solving the quantum maze 15

7 Summary 17

1 Introduction

Consider a quantum system of N q-bits whose states can be described by unit vectors in

-complex dimensional Hilbert space. The unit size of the sphere indicates that all of

the points are within distance of O(1) from each other if you were allowed to move along

geodesics on the unit sphere. Now imagine that you are only allowed to move in O(N

)

orthogonal directions out of O(2

). More precisely, at any point you are allowed to only

apply O(N) of one- q-bit gates or O(N

) of two- q-bit gates. Then the relevant question

is: what is the shortest distance connecting an arbitrary pair of points on the unit sphere?

This is like playing a very high-dimensional maze with a lot of walls and very few pathways.

There are two motivations to study the “quantum maze”: one computational and one

physical. First of all if we knew how to solve the “quantum maze” problem we would be

able to design the most eﬃcient quantum algorithms or in other words to construct the

shortest quantum circuits that can transform some simple initial state to the desired target

state. A problem which is known to be double exponentially hard O





(see ref. [1] for

a pedagogical discussion of computational complexities in context of quantum information

theory). The second reason has its roots in black-hole physics. It was conjectured that

black-holes are the fastest quantum computers in nature [2] and so in some sense the black-

holes know how to solve the “quantum maze”. Some other recent applications of the theory

of computational complexities in context of black-hole physics were discussed in ref. [3] and

in refs. [4, 5] in an attempt to tackle the ﬁrewall problem [6].

– 1 –

JHEP06(2016)001

In addition, a large body of work is directed towards establishing connections between

special kinds of quantum circuits (known as tensor networks) in context of the AdS/CFT

correspondence [7–10]. Indeed the tensor networks provide an interesting new prospective

on how the spacetime (on the AdS side) might emerge from a holographic state (on the

CFT side) by identifying the spacetime with a tensor network which would produce such

holographic state. In refs. [2, 11] the authors went even further and ﬁrst conjectured and

then veriﬁed that the complexity of the holographic CFT states are dual to the actions

over certain patches in the AdS spacetime.

In this paper we are going to expand and build upon the action-complexity conjec-

ture [2], but we shall not be concerned with transitions from only simple states to only

holographic states. Instead we will study transitions from an arbitrary initial |ψ

i to an

arbitrary ﬁnal |ψ

out

i state, i.e. the quantum maze problem. Nevertheless we will still con-

jecture that there must exist a (yet to be discovered) dual ﬁeld theory whose Euclidean

action describes the computational complexity of the smallest quantum circuit connecting

the two states, i.e.

C(|ψ

out

i, |ψ

i) = S

[Φ] (1.1)

where Φ is a collective notation for all ﬁelds. Note that both sides of eq. (1.1) are not

uniquely speciﬁed: the left hand side depends on size of Hilbert space and on the collection

of fundamental gates and the right hand side depends on the Lagrangian and on the region

of integration, but it is quite possible that for certain collections of fundamental gates there

exist a Lagrangian with desired properties.

The paper is organized as follows. In section 2 we deﬁne path integrals for the dual

theories of quantum computation and discuss the symmetries that the dual theory should

possess. In section 3 we argue that the dual theory can be conveniently described as a Klein-

Gordon lattice ﬁeld theory on an N + 1 dimensional space-time with geometrically ﬂat,

but topologically compact spatial slices. In section 4 we show how one can map classical

solutions on the dual theory side to quantum circuits on the quantum computation side.

In section 5 we identify the Z-problem of the Klein-Gordon theory which corresponds

to the massless excitations problem (Goldstone mode) and attempt to ﬁx it using Higgs

mechanism. In section 6 we discuss how the dual theory description can be used for solving

the “quantum maze” problem in sub-exponential time, but for that we must consider curved

spatial geometry and/or more complicated spatial topologies. In section 7 we summaries

the main results of the paper.

2 Path integrals for dual theories

Our initial task is to describe a possible construction of the dual ﬁeld theories for a system

of N q-bits with fundamental gates consisting of all one- q-bit and two q-bit operations.

Since the total number of degrees of freedom of such system is ﬁnite (dimensionality of

Hilbert space is only 2

), we are directed towards dual theories on the lattice with ﬁnite

fundamental domain, i.e.

C(|ψ

out

i, |ψ

i) =

(Φ

) , (2.1)

– 2 –

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