A compiler for universal photonic quantum computers 1stFelix Zilk

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A compiler for universal photonic quantum

computers

1st Felix Zilk

Christian Doppler Laboratory

for Photonic Quantum Computer

Faculty of Physics

University of Vienna

Vienna, Austria

felix.zilk@univie.ac.at

4thKarl F¨

urlinger

MNM-Team

Ludwig-Maximilians-

Universit¨

at (LMU)

Munich, Germany

fuerlinger@nm.iﬁ.lmu.de

2nd Korbinian Staudacher

MNM-Team

Ludwig-Maximilians-

Universit¨

at (LMU)

Munich, Germany

staudacher@nm.iﬁ.lmu.de

5th Dieter Kranzlm¨

uller

MNM-Team

Leibniz Supercomputing

Centre (LRZ)

Garching, Germany

dieter.kranzlmueller@lrz.de

3rd Tobias Guggemos

Christian Doppler Laboratory

for Photonic Quantum Computer

Faculty of Physics

University of Vienna

Vienna, Austria

tobias.guggemos@univie.ac.at

6th Philip Walther

Christian Doppler Laboratory

for Photonic Quantum Computer

Faculty of Physics

University of Vienna

Vienna, Austria

philip.walther@univie.ac.at

Abstract—Photons are a natural resource in quantum infor-

mation, and the last decade showed signiﬁcant progress in high-

quality single photon generation and detection. Furthermore,

photonic qubits are easy to manipulate and do not require partic-

ularly strongly sealed environments, making them an appealing

platform for quantum computing. With the one-way model, the

vision of a universal and large-scale quantum computer based

on photonics becomes feasible. In one-way computing, the input

state is not an initial product state |0i⊗n, but a so-called cluster

state. A series of measurements on the cluster state’s individual

qubits and their temporal order, together with a feed-forward

procedure, determine the quantum circuit to be executed. We

propose a pipeline to convert a QASM circuit into a graph

representation named measurement-graph (m-graph), that can be

directly translated to hardware instructions on an optical one-

way quantum computer. In addition, we optimize the graph using

ZX-Calculus before evaluating the execution on an experimental

discrete variable photonic platform.

Index Terms—Quantum Computing, Photonic QC, Measure-

ment Based QC, One-way QC, ZX-Calculus

I. INTRODUCTION

Photons are a natural candidate for quantum computing,

yet such systems are not very prevalent in Cloud or High-

Performance Computing (HPC) platforms. However, photonic

systems should be considered a valid competitor to other

platforms; recent ﬁndings show setups with up to 14 entangled

photons [1].

HPC is in an era of specialization, where an increasing num-

ber of accelerator devices are integrated in general-purpose

computing machines [2]. Quantum computers represent an

especially powerful type of accelerator, promising speed-

ups for unstructured search and combinatorial optimization

problems [3]–[5]. As quantum technology matures, it is im-

portant to enable integration of quantum processing units

(QPUs) in the HPC ecosystem and to support heterogeneous

programming that integrates classical and quantum aspects,

for example by means of ofﬂoading [6]–[11]. Here, quantum

algorithms are usually expressed as ofﬂoaded computational

kernels in a domain-speciﬁc language for the quantum circuit

model (e.g., QASM) [8], [12].

Although photons are an attractive platform for QPUs, a

direct translation of the quantum circuit model into photonic

components is impractical. Photonic two-qubit gates are in-

trinsically probabilistic; hence, an increasing number of gates

comes with an exponential decrease in the circuit’s success

probability. That is why measurement-based schemes [13]–

[15] are an appealing alternative, in particular the one-way

model of quantum computing [15]–[18]. Here, computation

is carried out solely by single-qubit measurements on highly-

entangled multipartite states – so-called cluster states [19].

The model is equivalent to the circuit model [20], but efﬁcient

methods for translation are still rare [21], [22].

Contribution: This paper describes our efforts to develop

a compiler for QASM kernels that targets discrete variable

photonic platforms. We propose a pipeline to translate from

QASM to a graph representation, named measurement-graph,

or m-graph. The m-graph is optimized with ZX-Calculus [23]

and mapped to hardware instructions for a photonic one-way

processor. This marks a ﬁrst step towards accessing photonic

QPUs with HPC systems.

II. BACKGROUND

The quantum circuit model performs computations by se-

quentially applying unitary gates to qubits in a quantum

product state |0i⊗n[4].

arXiv:2210.09251v1 [quant-ph] 17 Oct 2022

1 2 34 1 2

1 2

231 2

“Linear”

H1,3,4|GHZi

“Box”

“GHZ”

H1,4|ψi

l.c.

=l.c.

l.c.

Linear

GHZ

graph states

Fig. 1. Generation of the 4-qubit Linear and GHZ graph state from |ψi=1

2(|0000i+|0011i+|1100i − |1111i)and |GHZi=1

√2(|0000i+|1111i).

The graph states in each are local complementary (l.c.) with each-other, and can be use as input states for the one-way-model.

In contrast to the circuit model, the paradigm of quantum

annealing [24] bases on the unitary evolution of the underlying

system Hamiltonian.

A. The measurement-based one-way model

The one-way model, however, bases entirely on adaptive

single-qubit measurements that drive the computation [14],

[15], [25]. Here, the initial state of individual qubits is the

|+istate, and they are pairwise coupled via CZ operations to

form a graph state, which serves as the computational resource.

Then, single-qubit measurements on this resource state achieve

universal quantum computation.

Acluster state is a type of graph state in which the

underlying graph structure has the form of a two-dimensional

orthogonal grid. Graph states are highly-entangled multipartite

states, and we represent them mathematically as a graph

G(V, E), with vertices Vrepresenting physical qubits and

edges Eindicating entanglement between qubits. An arbitrary

graph state |Giof Vqubits and Eedges is [26]

|Gi=

Y

(a,b)∈E

CZa,b

O

v∈V|+iv(1)

Fig. 1 shows a collection of 4-qubit cluster states, arranged as a

two-dimensional lattice. The states are locally complementary

(l.c.) if one can be transformed into the other with single-qubit

transformations and SWAP operations only.

To carry out computation, we subsequentially measure on

connected physical qubits jin the equatorial basis Bj(α) =

{|αij,|−αij}, where |±αij= 1/√2(|0ij±eiα |1ij). Mea-

surements of physical qubits in the basis Bj(α)induce the

rotation HRz(−α)|ψion encoded logical qubits up to a

Pauli-Xcorrection (cf. Fig. 2). CZ gates are inherently built

into the computational resource state as links between two

qubits.

Thus, the native gate set in the one-way model may be

deﬁned as G={HRz(−α), CZ}, which is indeed universal.

Unlike the unitary evolution in the gate-based model, the

non-unitary action of measurement is irreversible. As each

measurement’s outcome is random, the desired result only

occurs in some cases. Hence, a feed-forward protocol [18]

compensates undesired results by adapting future measurement

bases according to earlier outcomes.

|ψiRz(−α)H

→Xm|ψ0i

|+i

One-Way-Model

→|ψ0i

→m={0,1}

Fig. 2. Circuit representation of the conceptual difference between the circuit

model with unitary transformation and the one-way model. Upper circuit:

shows the transformation of a qubit in state |ψiwith the unitary HRz(−α)

to state |ψ0i. Lower circuit: shows the same transformation with the one-way

model, where the new state |ψ0iis than teleported to the bottom qubit by

measuring the upper one (up to a Pauli-Xcorrection, that is based on the

measurements output m).

As an example, refer to the single-qubit computation in

Fig. 2. The result mof the upper wire measurement (red

cross mark) inﬂuences the Pauli-Xcorrection on the lower

output wire. If the outcome is m= 0, the algorithm works

as expected; however, if m= 1, a Pauli error is introduced

and corrected before the ﬁnal measurement. The cascaded

execution of this procedure allows for the implementation of

arbitrary single-qubit rotations. In fact, feed-forward control

makes one-way quantum computation deterministic.

Furthermore, any quantum circuit can be converted to a

measurement pattern on a sufﬁciently large cluster state [14],

[15], [25].

B. Implementation with a photonic processor

Photons are excellent candidates for building quantum com-

puters; they are easy to generate and detect, robust against

decoherence, and optical experiments can realize accurate

single-qubit gates easily. However, deterministic interactions

of two photons are experimentally impossible, and photonic

two-qubit gates are of probabilistic nature [15], [16].

In the one-way model, these nondeterministic operations

prepare the cluster state, just before any logical computation

takes place [15], [16].

Post-selection techniques ensure successful generation of

cluster states, such that it ignores certain detection events from

the results where the cluster state generation failed [17], [18].

High-precision measurements in an arbitrary basis are

achieved, for example, with phase retarders (wave plates)

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Acompilerforuniversalphotonicquantumcomputers1stFelixZilkChristianDopplerLaboratoryforPhotonicQuantumComputerFacultyofPhysicsUniversityofViennaVienna,Austriafelix.zilk@univie.ac.at4thKarlF¨urlingerMNM-TeamLudwig-Maximilians-Universit¨at(LMU)Munich,Germanyfuerlinger@nm.i.lmu.de2ndKorbinianStaudacher...

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