Holography as a resource for non-local quantum computation Kﬁr DolevaSam Creea

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Holography as a resource for non-local quantum

computation

Kﬁr DolevaSam Creea

aStanford Institute for Theoretical Physics, Stanford University, 382 Via Pueblo Mall, Stanford,

CA 94305-4060, U.S.A.

E-mail: Dolev@stanford.edu

Abstract: If two parties share suﬃcient entanglement, they are able to implement any

channel on a shared bipartite state via non-local quantum computation – a protocol consist-

ing of local operations and a single simultaneous round of quantum communication. Such

a protocol can occur in the AdS/CFT correspondence, with the two parties represented by

regions of the CFT, and the holographic state serving as a resource to provide the necessary

entanglement. This boundary non-local computation is dual to the local implementation of

a channel in the bulk AdS theory. Previous work on this phenomenon was obstructed by the

divergent entanglement between adjacent CFT regions, and tried to circumvent this issue

by assuming that certain regions are irrelevant. However, the absence of these regions intro-

duces violent phenomena that prevent the CFT from implementing the protocol. Instead,

we resolve the issue of divergent entanglement by using a ﬁnite-memory quantum simulation

of the CFT. We show that any ﬁnite-memory quantum system on a circular lattice yields

a protocol for non-local quantum computation. In the case of a quantum simulation of a

holographic CFT, we carefully show that this protocol implements the channel performed

by the local bulk dynamics. Under plausible physical assumptions about quantum compu-

tation in the bulk, our results imply that non-local quantum computation can be performed

for any polynomially complex unitary with a polynomial amount of entanglement. Finally,

we provide a concrete example of a holographic code whose bulk dynamics correspond to

a Cliﬀord gate, and use our results to show that this corresponds to a non-local quantum

computation protocol for this gate.

arXiv:2210.13500v1 [quant-ph] 24 Oct 2022

Contents

1 Introduction 1

2 Background 5

2.1 Non-local quantum computation 5

2.2 AdS/CFT 6

2.2.1 Entanglement wedge reconstruction 6

2.2.2 Causal structure discrepancy 7

3 Non-local computation in generic local (1 + 1)-dimensional systems 9

3.1 Many-body NLQC 9

3.2 Decomposing the boundary unitary 12

3.2.1 Decomposition via Cauchy slices 12

3.2.2 Decomposition via limited spread 12

4 Simulating holographic CFTs 14

4.1 Intuition & necessity of simulation features 15

4.2 Formal deﬁnition 16

4.3 Existence of the simulation 17

5 Non-local computation via holographic states 18

5.1 Applying many-body NLQC to a holographic simulation 19

5.2 Mapping the NLQC inputs/outputs to the boundary 22

5.3 Mapping the NLQC inputs/outputs to the simulation 22

5.4 Extracted protocol implements bulk dynamics 23

5.5 Error tracking 24

5.6 Entanglement consumption 25

6 Which computations are possible? 26

6.1 Constraints on bulk quantum computation 28

6.2 Polynomial entanglement for polynomially complex unitaries 29

7 A toy model 30

8 Relation to previous work 33

9 Discussion 36

9.1 Implications for Quantum Gravity 36

9.2 Implications for Quantum Information 37

9.2.1 Better NLQC protocols 37

9.2.2 Practical applications for holographic codes 37

9.2.3 Suboptimality of Bell pairs 38

9.3 Future directions 39

– i –

A Simulation error correction properties from correlation functions 40

A.1 Exact case 41

B Error from approximate correlation functions 42

B.1 Approximate code 42

B.2 Encoding error 46

B.3 Decoding error 47

B.4 Dynamical duality error 48

C Error from imperfect light-cone 49

C.1 Unitary decomposition 49

D Proof that entanglement can be conﬁned to Xregions 49

1 Introduction

In position-based cryptography [1–5], individuals use their spacetime position as crypto-

graphic credentials. In the simplest task, position veriﬁcation, a prover must prove to a

veriﬁer that they are in a speciﬁc spacetime region. They do so by performing a local com-

putation on signals sent by the veriﬁer, and returning the outputs back to the veriﬁer. The

protocol is designed so that only someone in the authorized location is capable of locally

performing the computation without violating causality. This may be used, for example,

to ensure that only someone inside a trusted facility is able to read a message. However,

two dishonest provers can collaborate to non-locally simulate this local computation using

non-local quantum computation (NLQC) – a quantum task in which two parties implement

a joint channel on the systems they hold when limited to one round of communication [2–22]

– provided they share enough resources to do so.

The question thus remains to characterize exactly how resource-intensive NLQC is.

Only suﬃciently simple computations are relevant for position-based cryptography, since a

computation that is too complex cannot be performed in time even by an honest prover

without jeopardizing the causality constraints. Thus if all low-complexity computations

can be eﬃciently performed non-locally – i.e. with only a polynomial amount of resources

– then practical, secure position veriﬁcation is impossible.

There exist some partial results about characterizing the resource requirements for

NLQC, which focus on quantifying the number of Bell pairs required. The tightest known

resource requirement upper bound for general unitaries was derived in Ref. [23] by giving

an explicit general purpose protocol. This protocol consumes a number of Bell pairs expo-

nential in the total number of qubits on which the unitary acts. More eﬃcient protocols

can sometimes be found by exploiting the structure of the unitary [6,24] or restricting it

to a particular class [16,25]. Linear lower bounds are known for speciﬁc tasks [26–28]. For

some unitaries, a loglog lower bound in terms of complexity is given in Ref. [27].

– 1 –

The anti-de Sitter/conformal ﬁeld theory (AdS/CFT) correspondence [29,30] is a fam-

ily of holographic dualities [31,32] that relate a bulk theory of quantum gravity in asymp-

totically AdS spacetime to a CFT living on its boundary. A notable pattern in these

dualities is that information-theoretic quantities in the boundary are related to geometric

quantities in the bulk [33–37]. It was recently argued that the boundary dynamics of certain

holographic systems can be interpreted as executing an NLQC protocol that implements

the bulk dynamics [38]. This observation has yielded a number of results and conjectures

about both AdS/CFT and NLQC [27,35,38], such as a method of placing constraints on

bulk dynamics, and a tension between the existence of universal quantum computers in

holography and the possibility of secure position-based cryptography. The tension arises

from the claim that if such a computer could be placed in the bulk, the boundary could

perform any unitary non-locally with an amount of entanglement scaling at most polyno-

mially with the complexity. This would dramatically improve upon the general purpose

protocol of Ref. [23] for simple unitaries. If this is demonstrated rigorously, then all simple

unitaries could be eﬃciently implemented non-locally, and thus secure position veriﬁcation

would be impossible. However, it remains to establish a precise connection between this

behavior and the task of NLQC, as previous attempts have been non-rigorous, and relied

on signiﬁcant assumptions whose validity we question in this work.

Here we provide a more careful and detailed demonstration of this connection without

relying on those assumptions. We formally establish the connection between holography

and NLQC by carefully showing that it is possible to extract a protocol from a simulation of

the boundary CFT that implements the channel associated with the local bulk dynamics.

The simulation only needs to accurately capture simple correlation functions of certain

operators, preserve locality of operators, and satisfy an approximate light-cone, in ways

that we make precise. Our protocol depends only on the initial CFT simulation state

and its Hamiltonian, ensuring that it captures the particular mechanism that AdS/CFT

uses to accomplish the task; this is in contrast to the suggestion proposed in Ref. [35],

which we show in Section 8requires use of operations not performed by the CFT itself. It

also explicitly uses ﬁnite-memory1quantum systems, rather than the inﬁnite-dimensional

systems associated with regions of a ﬁeld theory.

We start by introducing a general-purpose “many-body” NLQC protocol for any ﬁnite

memory (1 + 1)-dimensional quantum system living on a circular lattice. It uses an initial

state of the system as the resource, and implements some computation using the local

dynamics of the system. Such systems mimic the causal structure of a (1 + 1)-dimensional

CFT since the locality of the Hamiltonian gives rise to an emergent light-cone described

by the Lieb-Robinson velocity [39]. When this protocol is applied to a quantum system

simulating the relevant features of a holographic CFT, the computation that is implemented

is exactly the one corresponding to the bulk dynamics of interest (see Fig. 1).

It is not clear how to simulate a CFT with a (1 + 1)-dimensional system suitable for

NLQC protocol extraction, such that the underlying computation is the same. A perfect

simulation of a ﬁeld theory with a ﬁnite-dimensional system is impossible, because subre-

1i.e. a ﬁnite number of qubits

– 2 –

gions of quantum ﬁeld theories have divergent entropy. Fortunately, any CFT is renormal-

izable, meaning that only a ﬁnite subset of its degrees of freedom are ever relevant to a

particular phenomena. Thus in order to preserve the NLQC performed by a holographic

CFT, a simulation of it only needs to capture the subset of degrees of freedom relevant

to the bulk computation, and their relevant dynamics. Thus we look for a minimal set of

CFT features the simulation must reproduce in order for the associated protocol to imple-

ment the bulk dynamics. We ﬁnd these boil down to 1. preserving the locality structure

of the CFT, 2. reproducing certain low-order correlation functions, and either 3a. an ap-

proximate light cone is satisﬁed even for operators without counterparts in the CFT (e.g.

due to a Lieb-Robinson velocity [39]), or 3b. Cauchy slices2can be locally deformed. The

correlation functions are generally captured only inexactly, up to some error quantiﬁed by

a dimensionless parameter δ. The approximate light-cone means that the commutator of

two operators is bounded proportional to aexp(−b(d−t)), where dand tare the space

and time separations of the operators, and aand bare error parameters. As the resolution

of the simulation is increased, bshould diverge to inﬁnity so that the light-cone becomes

exact, and ashould not increase so quickly that the bound becomes trivial in that limit.

We argue for the existence of such a simulation of a holographic CFT by looking at

known examples of other CFT simulations. In particular, the simulation method of Ref. [40]

makes use of a rigorous deﬁnition of a CFT using a technique known as operator-algebraic

renormalization [41–43], and captures at least features 1, 2 and 3b. On the other hand,

Ref. [44] considers a lattice system whose low energy limit reproduces a continuum U(1)

gauge theory in a way that captures features 1,2, and 3a suﬃciently well. No rigorous

results are known about simulating strongly coupled QFTs (such as holographic CFTs) due

to technical challenges, but it is widely believed that similar techniques should apply to

them.

Our main result is the following. Let ULbe a channel representing the bulk dynamics

that we construct an NLQC protocol for. Alice and Bob will use an encoding channel E0to

input their respective systems into the holographic simulation. We show that they can then

apply an approximation Vsim of the local time evolution in the delocalized form required

for NLQC, i.e. using local operations and one round of communication. Finally, they use a

recovery channel Rτto decode their output systems from the simulation. Aside from the

simulation’s error parameters δ,aand b, there will also be error arising from the inexactness

of the semiclassical description of the bulk. The emergence of the bulk is only exact in the

limit of weak gravity, that is, as Newton’s constant GN→0. We show that the error in

the application of ULgoes as

||Rτ◦ Vsim ◦ E0− UL||≤cCF T pGN+csim√δ+cspreadaexp(−b∆τ),

where cCF T ,cspread,csim and ∆τare O(1) positive parameters, and GNis the Newton’s

constant of the holographic theory. All these errors can be made arbitrarily small by

increasing the resolution scale of the simulation, and decreasing Newton’s constant; however

these will both be associated with increases in the amount of entanglement required. Loosely

2A Cauchy slice is a surface which every time like curve without end points crosses exactly once.

– 3 –

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Holographyasaresourcefornon-localquantumcomputationKrDolevaSamCreeaaStanfordInstituteforTheoreticalPhysics,StanfordUniversity,382ViaPuebloMall,Stanford,CA94305-4060,U.S.A.E-mail:Dolev@stanford.eduAbstract:Iftwopartiessharesuciententanglement,theyareabletoimplementanychannelonasharedbipartitestatev...

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