Design Constraints for Unruh-DeWitt Quantum Computers. Eric W. Aspling1John A. Marohn2 3and Michael J. Lawler1 4 5 1Department of Physics Applied Physics and Astronomy Binghamton University Binghamton NY 13902_2

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Design Constraints for Unruh-DeWitt Quantum Computers.

Eric W. Aspling,1John A. Marohn,2, 3 and Michael J. Lawler1, 4, 5

1Department of Physics, Applied Physics, and Astronomy, Binghamton University, Binghamton, NY 13902

2Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14853

3Department of Materials Science and Engineering, Cornell University, Ithaca, NY 14853

4Department of Physics, Cornell University, Ithaca, NY 14853

5Department of Physics, Harvard University, Cambridge, MA 02138

(Dated: April 8, 2024)

The Unruh-DeWitt particle detector model has found success in demonstrating quantum information channels

with non-zero channel capacity between qubits and quantum ﬁelds. These detector models provide the necessary

framework for experimentally realizable Unruh-DeWitt quantum computers with near-perfect channel capacity.

We propose spin-qubits with gate-controlled coupling to Luttinger liquids as a laboratory setting for Unruh-

DeWitt detectors and explore general design constraints that underpin their feasibility in this and other settings.

We present several experimental scenarios including graphene ribbons, edges states in the quantum spin Hall

phase of HgTe quantum wells, and the recently discovered quantum anomalous Hall phase in transition metal

dichalcogenides. Theoretically, through bosonization, we show that Unruh-DeWitt detectors can carry out quan-

tum computations and identify when they can make perfect quantum communication channels between qubits

via the Luttinger liquid. Our results point the way toward an all-to-all connected solid state quantum computer

and the experimental study of quantum information in quantum ﬁelds via condensed matter physics.

I. INTRODUCTION

Unruh-DeWitt (UDW) detectors originated as a thought ex-

periment by Unruh [1] (later extended by DeWitt [2]) to model

an accelerating qubit in a vacuum. Unruh showed that an ac-

celerating observer would view the ground state of a quantum

ﬁeld as a mixed state and see a loss of quantum information

hidden behind the Killing horizon [3]. Today, there are paral-

lel efforts to utilize UDW detectors to advance science in cos-

mology, high energy physics, and condensed matter physics.

Cosmologists use them to model information in highly accel-

erated frames of reference (e.g. in and around black holes),

high energy theorists use them without acceleration to study

quantum information ﬂow via quantum ﬁelds, a ﬁeld called

relativistic quantum information [4–10]. Condensed matter

experimentalists use UDW detectors such as nitrogen-vacancy

centers and superconducting interference devices, calling the

detectors “quantum sensors”, to sensitively detect electromag-

netic ﬁelds produced by a wide variety of systems from quan-

tum materials to systems outside of condensed matter like can-

cer cells. But currently, experimentalists demand much less

from the UDW detectors than theorists, having yet to use them

to study the ﬂow of quantum information in complex systems.

A feature that theorists require of UDW detectors is the

ability to turn their coupling to their environment on and off

rapidly. Consider a spin qubit coupled to a one-dimensional

wire modeled as a Kondo-like impurity. A simple look at such

a quantum computer is displayed in Fig. 1 which showcases

natural scalability as a feature that preserves all-to-all quan-

tum communication. A setup such as this is capable of sensing

the ﬂow of small currents in the wire. Turning the Kondo-like

coupling on and off rapidly, however, turns it into an emit-

ter that sends signals through the wire, signals to be picked

up by another such spin qubit acting as a detector. The net

result of this communication amounts to a quantum gate that

acts unitarily on the combined qubit-wire system. In Fig. 7,

we take this idea to the next level: the design of an all-to-all

connected solid-state quantum computer where gates can be

applied to distant qubits enabled by communication via quan-

tum coherent wires. The possibility that a quantum wire could

achieve such communication dates back to at least as early

as 2007 [11, 12]. Control over timing, therefore, enables the

UDW detector to emit and receive quantum information. This

propagation of quantum information offers clear beneﬁts to

quantum technology.

In addition to practical applications in quantum comput-

ing and communication, timing-controllable UDW detectors

would allow the study of complex quantum systems in a new

way. Careful construction of quantum information channels

through these systems allows for a deeper understanding of

their quantum properties without directly carrying out projec-

tive measurements on the system or inferring them through

measurements of expectation values. Simulating channel ca-

pacity, for example, would show how entanglement spreads

within them and how quantum information becomes scram-

bled. If we could similarly study physical systems, we could

directly probe their entanglement properties.

Studying quantum information channels into and out of

complex quantum systems provided by UDW detectors will

also enable these systems to become part of quantum tech-

nology. For example, a system described by a quantum

ﬁeld could become a component in a quantum computer that

can carry out computations (these ﬁelds are known as ﬂying

qubits). The grand application of such a computer would then

be to simulate quantum ﬁeld theory, a task long predicted to be

a consequence of quantum computing [13–16]. Such a sim-

ulator, for example, could simulate Dirac fermions directly

without needing to overcome the fermion doubling problem

caused by discretization.

One possible system to achieve simulation of Dirac

fermions is known as the Yao-Kivelson model, which has been

studied in Ref. 17. The Yao-Kivelson model provides a solv-

able model of topological edge states that can be expressed as

arXiv:2210.12552v3 [quant-ph] 5 Apr 2024

(a) Qubit A can access all

qubits on the system via

left- and right-movers.

(b) Opening up qubits will

cause interactions with the

left- and right-movers.

qubits restricts of our left-

and right-movers to paths

(1) or (2).

(d) Opening the bulk

between the blocks gives

Qubits B and C direct

access to the entirety of the

qubits.

FIG. 1: Figures (a-d) show a simplistic view of our quantum

bus. Qubits (such as qubits A and B) are placed around the

edges of a Luttinger liquid. The potential wall is lowered, as

in Fig. (b) and the qubits are able to interact with the

topological edge states. Figures (c) and (d) indicate how

scaling up this system can be done easily by adjusting where

the bulk of our ﬁelds lives through raising and lowering the

potential barrier.

Majorana fermions. Dangling qubits near the edge state pro-

vide a comparable model to a UDW detector and offer helpful

insights regarding restrictions to our system. In Sec. II B we

discuss this system as it pertains to our own in more detail. In

the near term, we expect a UDW quantum computer that uti-

lizes a quantum bus like that of Fig. 1 to be better equipped at

enhancing error-correcting codes by exploiting all-to-all con-

nectivity. So we see that timing-controlled UDW detectors

will have both fundamental and technological applications.

In this paper we propose and assess three potential sys-

tems for realizing the study of quantum information ﬂow via

timing-controlled UDW detectors coupled to quantum mate-

rials: HgTe/CdTe heterostructures in the quantum spin Hall

phase [18, 19], graphene spin qubits [20–24], and Moir´

e tran-

sition metal dichalcogenides (TMDs), in either the anomalous

or spin quantum-Hall regime [25, 26]. Each of these systems

contains spin qubits coupled to Luttinger liquids. We begin

with a theoretical analysis combining the formalism of UDW

detectors with the bosonization of Luttinger liquids. This

innovation gives us the ability to engineer new systems for

exploring quantum information ﬂow through quantum ﬁelds.

Using the results of Simidzija et al. [5–7], we show that a

non-zero channel capacity should exist in these systems. We

provide a library of Hamiltonians that characterize the qubit-

ﬁeld quantum transduction constraints demonstrated in this

paper, bolstering the natural viability of quantum electron-

ics/communication. We assess the experimental viability of

the three proposed systems. We close by discussing future the-

oretical work, outlining the many new avenues of information

research generated by the timing-controlled UDW detectors

proposed here.

II. QUANTUM INFORMATION FLOW IN QUANTUM

MATERIALS

The trend for scaling up quantum computing consists of

larger and larger numbers of qubits carrying out linear op-

erations over longer length scales. This approach at scal-

ing seems natural as error correcting codes are more accurate

with more qubits [27–30]. However, topological systems such

as the fractional quantum Hall effect could provide a quan-

tum bus of all-to-all connected qubits which offer a robust er-

ror correcting code [13] that scales at least like Fig. 1. The

all-to-all communication channels provide situational error-

correcting codes based on stabilizer codes [31] and the peri-

odic condition of our quantum bus resolve length scales. The

most important question is thus: how well does our system

process quantum information? Undertaking this task involves

devising quantum channels displaying maximal channel ca-

pacity.

A. Devising the Quantum Channel

Quantum channels present the necessary formalism to an-

alyze entanglement propagation through a quantum circuit

[32, 33]. Recently, high-energy physicists have made progress

investigating quantum channels as they exist between ﬁelds

and qubits. As mentioned in the introduction, they use UDW

detector formalism that utilizes a smearing function to spread

a two-dimensional Hilbert space onto an inﬁnite-dimensional

space. This formalism carries a series of complications such

as limitations due to the no-cloning theorem and informa-

tion spreading due to Huygen’s principle in spatial dimensions

higher than one [5, 34]. In this regard, an analogy to classi-

cal wireless communication is not possible. Instead, advanc-

ing quantum devices such as quantum wires may prove more

proﬁtable.

Using quantum gate formalism, we evaluate the quantum

coherent information, a ﬁgure of merit for the channel, which

is analogous to mutual information in classical information

theory. For the systems discussed in this paper, quantum co-

herent information is computed according to the circuit shown

in 2. The construction of candidate unitaries, such as UA

and UBin Fig. 2, needed for our quantum channel, is done

carefully in Sec. III. For now, we will outline some design

constraints of these unitaries that enable them to successfully

transport quantum information.

Simidzija et al. as well as others, have recently laid the

groundwork for models that successfully outline the neces-

Input Output

Noisy Channel

Qubit C

Qubit A

Field

Qubit B

|ψCA⟩

|ψCB ⟩

|S⟩A

ϕInternal Interactions

ϕ′

|0⟩B

tAtB

FIG. 2: Quantum coherent information is a measure of

quantum information ﬂowing through a quantum channel. To

compute it, we use the above circuit diagram. Initially, Qubit

A is entangled with Qubit C in a Bell state. Then Qubit A

coupled to the Field φthrough UAand later Qubit B |0⟩Bis

similarly coupled through UB. Finally, the entanglement is

measured between Qubit B and Qubit C. If these qubits are

entangled, then the coherent information is positive and

quantum information ﬂowed through the noisy channel.

sary conditions of just such a channel [5–7, 35]. They have

shown that UDW unitaries which behave as controlled uni-

tary gates, lead to entanglement-breaking channels when pro-

cessed alone. However, carefully applying two of these rank-

one unitaries breaks the controlled-gate structure of the cir-

cuit, allowing them to properly encode (or decode) informa-

tion from a spin structure onto a scalar bosonic ﬁeld. In other

words, the channel created by gates with these unitaries can

have a positive-valued coherent information that scales with

coupling and smearing parameters.

More elusive is a schematic for strongly coupled fermionic

systems using UDW detectors. As we will demonstrate,

the formalism describing quantum channels, traces over the

ﬁeld and results with a correlator of ﬁeld operators. Map-

ping fermions to bosons through bosonization has an equiva-

lence at the level of the correlators. We ﬁnd that through the

bosonization of our Luttinger liquid, we can create different

approaches to solve this problem. Section III aims to provide

a library of these gates which enable channels with non-zero

capacity. Furthermore, we claim that careful parameter selec-

tion can theoretically create a near-perfect quantum channel.

B. Design parameters governing channel performance

There are many parameters governing a ﬁeld-mediated

quantum channel between two qubits. We can separate the

channel into two components, gates between the qubit and

ﬁeld and the propagation pathway the quantum information

travels along within the ﬁeld itself.

Generally, a gate between a qubit and a ﬁeld is governed

by a coupling function J(x,t). We can break this function into

three factors as is common in the relativistic quantum infor-

mation literature. One is a switching function χ(t)normal-

ized to Rdt χ(t) = 1 that turns the gate on and off. Another is

a smearing function p(x,y)that couples a qubit at xto the ﬁeld

at various points y. It too is normalized with Rdy p(x,y) = 1.

Ideally, both χ(t)and p(x,y)are non-negative functions. Pre-

sumably, p(x,y)is non-zero only inside the quantum dot

(Qdot) hosting the qubit. Finally, there is the overall dimen-

sionless strength of the coupling J. Hence the coupling func-

tion J(x,t)is naturally parameterized by this strength J, a

switching time tsw and a smearing length λs.

For our models in Sec. III, we use a Dirac delta-like switch-

ing function with tsw =0, a mathematically convenient but

physically impossible situation. It allows the gate to produce

a change in the ﬁeld that remains perfectly localized within

the smearing length. For tsw >0, during the action of the gate

information will spread away from the qubit at the velocity

v, the effective speed of light governing the relativistic ﬁeld.

Hence, if we have tsw <λs/v, the effect of the gate will remain

nearly localized within the smearing length, and a physically

realizable gate will behave similarly to our idealized gates.

Given λs,tsw, and v, we now have two dimensionless pa-

rameters governing the design of a gate: the coupling strength

Jand the gate-localization quality Qloc =tswv/λs. A small

Qloc is a design constraint for a UDW quantum computer. If

it is too large, quantum information will spread over large dis-

tances and during the action of the gate making, it is difﬁcult

to recapture. A small J, however, implies the gate has little ef-

fect. Hence for good channel performance, we will want gates

with a small Qloc and large J.

Earlier studies, Refs. 17, 36 identify another design con-

straint of quantum information channels in condensed matter

physics. Though different from an UDW Quantum Computer,

Refs. 17, 36 provide an approach to carry out such a process

by modeling a dangling qubit near a topologically protected

edge state or an end spin of a spin chain. Their perturba-

tive approach offers no analytic limits of a perfect quantum

information channel (a proof can be found in the discussion

surrounding Eq. 16 of Ref. 5). However, it offers other in-

sights, including identifying internal interactions that promote

scrambling. This study therefore highlights an important de-

sign criteria: to study quantum information in a quantum ma-

terial, it must ﬂow and be picked up within the scrambling

time tsof the material or it will be lost.

We therefore need to understand how, as the information

propagates, it is subject to scrambling by interaction effects

[37–39]. Measurements on the target qubit may detect the on-

set of quantum chaos caused by the system’s inherent disorder

[37, 40, 41]. Similarly, if the information runs into a magnetic

impurity acting like an uncontrolled qubit, it may be stolen by

it and never reach the intended qubit [42]. The information

could also be taken away by phonons and spread throughout

the host material [43]. So this intermediate stage is simultane-

ously an opportunity to study the physics of the host quantum

material at a quantum information level and a constraint on

the performance of the communication —it limits the distance

between communicating qubits to v/ts.

Given the above design constraints, we next turn to the

question; in ideal circumstances, does a perfect communica-

tion channel exist for qubits coupled to Luttinger liquids?

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DesignConstraintsforUnruh-DeWittQuantumComputers.EricW.Aspling,1JohnA.Marohn,2,3andMichaelJ.Lawler1,4,51DepartmentofPhysics,AppliedPhysics,andAstronomy,BinghamtonUniversity,Binghamton,NY139022DepartmentofChemistryandChemicalBiology,CornellUniversity,Ithaca,NY148533DepartmentofMaterialsScienceandEngi...

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Design Constraints for Unruh-DeWitt Quantum Computers. Eric W. Aspling1John A. Marohn2 3and Michael J. Lawler1 4 5 1Department of Physics Applied Physics and Astronomy Binghamton University Binghamton NY 13902_2

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