Parallelizable Synthesis of Arbitrary Single-Qubit Gates with Linear Optics and Time-Frequency Encoding Antoine Henry1 2Ravi Raghunathan1Guillaume Ricard1Baptiste Lefaucher1

2025-05-02 0 0 858.4KB 21 页 10玖币
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Parallelizable Synthesis of Arbitrary Single-Qubit Gates
with Linear Optics and Time-Frequency Encoding
Antoine Henry,1, 2 Ravi Raghunathan,1Guillaume Ricard,1Baptiste Lefaucher,1
Filippo Miatto,1Nadia Belabas,2Isabelle Zaquine,1and Romain All´eaume1
1T´el´ecom Paris-LTCI, Institut Polytechnique de Paris,
19 Place Marguerite Perey, 91120 Palaiseau, France
2Centre for Nanosciences and Nanotechnology, CNRS, Universit´e Paris-Saclay,
UMR 9001,10 Boulevard Thomas Gobert, 91120 Palaiseau, France
We propose novel methods for the exact synthesis of single-qubit unitaries with high success
probability and gate fidelity, considering both time-bin and frequency-bin encodings. The proposed
schemes are experimentally implementable with a spectral linear-optical quantum computation (S-
LOQC) platform, composed of electro-optic phase modulators and phase-only programmable filters
(pulse shapers).
We assess the performances in terms of fidelity and probability of the two simplest 3-components
configurations for arbitrary gate generation in both encodings and give an exact analytical solution
for the synthesis of an arbitrary single-qubit unitary in the time-bin encoding, using a single-tone
Radio Frequency (RF) driving of the EOMs. We further investigate the parallelization of arbitrary
single-qubit gates over multiple qubits with a compact experimental setup, both for spectral and
temporal encodings. We systematically evaluate and discuss the impact of the RF bandwidth -
that conditions the number of tones driving the modulators - and of the choice of encoding for
different targeted gates. We moreover quantify the number of high fidelity Hadamard gates that
can be synthesized in parallel, with minimal and increasing resources in terms of driving RF tones
in a realistic system. Our analysis positions spectral S-LOQC as a promising platform to conduct
massively parallel single qubit operations, with potential applications to quantum metrology and
quantum tomography.
I. INTRODUCTION
Quantum information processing with light is a promising direction for near-term quantum computing. Quantum
photonics systems are currently placed at the forefront of the technological race to engineer well-controlled optical
quantum states in a Hilbert space of very high dimensionality, with a record value of 1043[1]. Another key advantage
of quantum photonics hardware is scalability based on photonic integration, now enabling compact photonic circuits
approaching 1,000 components for millimetre-scale footprints [2].
A drawback of using light is however that photons do not interact with one another, which means that conditional
operations need to be mediated by matter via some non-linear process [3]. Even though non-linear processes can
theoretically implement the desired operations, this can lead in practice to very inefficient schemes, i.e. with very
low probability of success. Linear Optics Quantum Computation (LOQC), introduced in the pioneering work of
Knill, Laflamme and Milburn in 2001 [4], performs universal quantum computing with light, using only linear optics
and postselection. A different scheme, spectral linear-optical quantum computation (S-LOQC), was proposed [5]
as a highly promising approach for scalable quantum information processing. S-LOQC harnesses photonic qubits
encoded over spectral modes (dual-rail frequency encoding) and gate transformations implemented with off-the-shelf
telecom components: Electro-Optics Modulators (EOM) and phase-only programmable filters (PS). S-LOQC notably
leverages the spectral degrees of freedom that can allow to reach high dimensionality much more easily than spatial
and polarization degrees of freedom. It can be used as a resource for parallelization of single qubit transformation, or
to build high dimensional quantum states (qudits).
The question of unitary gate synthesis in S-LOQC has already been studied in a series of work over the past few
years. In the seminal paper [5], a set-up comprising 2 PS [P] and 2 EOM [P] in the sequence [PEPE] was first
considered. Using numerical optimization techniques, a deterministic Hadamard gate for spectral dual-rail encoding
was theoretically designed with a unity fidelity and success. The question of parallelization was also studied, however
it was found that a minimum of 6 modes separating two qubits was needed to ensure a success probability >90%.
The number of components, ancilla bits and guard bands required were found to be higher for the synthesis of the
CZ-gate.
In [6], the authors adapted their scheme from [5] to experimentally implement high-fidelity frequency beamsplitters
and tritters with classical light. Two important modifications from their previous proposal involved reducing the
number of components from 4 to 3 (in an [EPE] sequence), and the use of phase-shifted sinewaves as the EOM RF-
driving as opposed to arbitrary waveforms, which place greater demands on bandwidth. Using this [EPE]-single-tone-
scheme, a beamsplitter was experimentally realized with fidelity of ≈0.99998 and success probability ≈97.39% at
arXiv:2210.11830v1 [quant-ph] 21 Oct 2022
2
1545.04 nm. The scheme was also shown to be well-suited for parallelization, where the separation between adjacent
beamsplitters was found to be a minimum of 4 spectral modes, allowing for 33 beamsplitters to be implemented
in parallel under the constraints of the experiment. Furthermore, with the addition of an extra tone to the EOM
RF-driving, a frequency tritter with fidelity ≈0.9989 and success probability ≈97.30% was synthesized.
The same [EPE] scheme was then used to implement distinct quantum gates (Hadamard and Identity) in parallel, in
frequency-bin encoding [7]. The corresponding programmable unitary was moreover used to tune the overlap between
adjacent spectral bins, which allowed to observe spectral Hong-Ou-Mandel interference with a visibility of 97%. More
generally, it was shown that individual single-qubit gate operations can be applied in parallel to each of an ensemble
of co-propagating qubits, where each operation can be smoothly tuned between the Identity and Hadamard gates.
More recently arbitrary control of spectral qubits was reported in [8] with the same set of components [EPE], (i)
experimentally with a single-tone sinewave modulation, and (ii) numerically for dual-tone modulation. The 2-qubit
CNOT gate with the same setup ([EPE] and BFC source) [9] was also reported. Gate reconstruction was performed
from measurements in the two-photon computational basis alone, and a fidelity F ≈ 0.91 was inferred.
In our work we explore S-LOQC further, with a focus on single qubit unitary gate synthesis, first for a single
qubit, and then for many qubits in parallel. S-LOQC is indeed naturally adapted for parallelization allowing, without
substantial change in the implementation, to apply the same unitary gate to many qubits in parallel. While being
used in seveval quantum information protocols [10,11] time-bin encoding has not been as much explored as frequency
encoding in the context of the Spectral LOQC platform. It has however been proved to be efficient for several
tasks in quantum information processing [12–15]. Similarly to frequency encoding, it offers the advantages of a high
dimensional accessible Hilbert space and we show here that time encoding allows for more expressive gate synthesis
possibilities than frequency encoding while using the same number of phase modulators and pulse-shapers components.
We investigate both [EPE] and [PEP] configurations of components, and show that both configurations can be used
to efficiently perform any unitary transformation, with unity fidelity and success probability. In particular we exhibit
an analytical solution for the synthesis of an arbitrary single-qubit unitary, using minimal ressources, i.e. a single-tone
Radio Frequency (RF) driving for the EOMs. Moreover we also show that those transformations can be applied in
parallel with time encoding to a larger number of qubits than what is possible with frequency encoding. We study,
for several gates, the trade-off between the quality (fidelity, success probability) of the synthesis, the parallelization
that can be achieved, and the ressources needed, in particular in terms of RF bandwidth.
The article is organized as follows: we introduce in section II the formalism for encoding both in time or frequency
basis and for the corresponding Hilbert spaces, and the description of the components in both basis. Section III
defines the general problem of qubit unitary gate synthesis that we tackle in this article and explains the rationale
of our approach. We notably detail the ”two-scattering model for the pulse-shaper in the time basis, that will be
instrumental for the new results reported in section IV. We finally detail how the performance parameters for gate
synthesis, namely fidelity and success probability are defined. The next two sections report our results, that consist in
a systematic and wide exploration of the different options, in terms of encoding bases, gates, and configurations (either
[EPE] or [PEP]). Section IV presents our results related to the synthesis unitary gates, for a single qubit. Remarkably,
we exhibit, in the time basis, an exact solution allowing the synthesis of an arbitrary single-qubit unitary with single
tone RF driving of the EOM, in both [EPE] and [PEP] configurations. These results are put in perspective with
single qubit unitary gate synthesis in the frequency basis, whose fidelity and success probability depend on the type
of gate that is targeted. Section V then presents our results related to the parallel synthesis of the same qubit gate
over many different qubits. Here again, we compare the two encodings, as well as the two considered configurations,
and discuss the interplay between the performance of the synthesis and the number of RF tones.
II. TIME AND FREQUENCY FORMALISM FOR S-LOQC
In this section we define the photonic states that we will employ and the theoretical description of the Pulse Shaper
(PS) and the Electro-Optic Phase Modulator (EOM). The descriptions of the devices must take into account the
EOM and PS physical characteristics, as these characteristics set a limit on the total number of available modes for
quantum manipulation.
A. Optical modes and quantum states
The system that we consider is composed of Moptical modes identified either by a frequency bin of width δω,
centered on ωjor by a time bin of width δt, centered on tk. Frequency modes |ωjiare linear combinations of time-bin
modes |tki. These two sets of basis vectors are connected by a discrete Fourier transform
3
|tki ≈ 1
√M
M−1
X
j=0
exp i2π
Mjk|ωji,(1)
|ωji ≈ 1
√M
M−1
X
k=0
exp −i2π
Mjk|tki.(2)
The interchange between the frequency and time bases is exact only in the continuous case, when M→ ∞ and
δω, δt →0.
To operate on single qubits, we divide our Hilbert space of dimension Minto M/2 independent subspaces according
to a choice of qubit encoding either in the frequency domain Hω
jusing two contiguous frequency bins or in the time
domain Ht
kusing modes separated by M/2 time bins, with j, k ∈[0, M/2−1].
Hω
j={|ω2ji,|ω2j+1i}, ω2j+1 −ω2j=δω, (3)
Ht
k=|tki,|tk+M/2i, tk+M/2−tk=M
2δt, (4)
Such a qubit encoding choice is made to take into account the technical limitations of the devices. In the present
case, the fact that the EOM couples mainly neighboring frequency modes justifies the choice of adjacent modes to
encode frequency-bin qubits. Conversely, the fact that the PS couples time modes that are M/2 time bins appart (cf
Appendix B) justifies the choice of encoding used for the time-bin qubits.
As the considered Hilbert spaces are orthogonal, we can write the sum of the sub-spaces as
H=M
jHω
j=M
kHt
k.(5)
We now introduce the devices that we are using for photonic qubit processing.
B. Pulse Shaper
The phase only programmable filters (PS) can shift the phase of frequency modes concurrently and independently.
It is therefore characterized by a set of Mangles {ϕj}(one angle per mode) that can be chosen freely, without
constraints. In general, the PS acts as the tensor product of distinct phase shifts operators, one per frequency mode
and its operator ˆ
UP S can be written
ˆ
UP S =
M−1
O
j=0
exp(iϕjˆ
Nωj),(6)
where ˆ
Nωjis the number operator of the ωjfrequency mode: ˆ
Nωj=Pnn|nωjihnωj|. However in the restricted case
of a single photon, we can thus treat the PS as a diagonal matrix P in the frequency basis. Its action on the 1-photon
subspace is described in Fig. 1.b.
P|ωji=eiϕj|ωji.(7)
In order to describe the action of the PS in the time basis (see Fig. 2), we use the Fourier transform ˜
P=FPF†
where the matrix Fand the detailed calculations are given in appendix Aand we obtain
˜
P|tki=1
M
M−1
X
k0=0
M−1
X
j=0
exp i2π
M(k0−k)j+ϕj|tk0i.(8)
The PS is therefore a scatterer in the time basis, coupling a single time mode to all the other modes. The number
of modes over which a single mode can be scattered will be an important optimization parameter, as shown in section
III C 1.
4
FIG. 1: Action of a. the Electro-optic Phase Modulator and b. the Pulse Shaper in the frequency basis
C. Electro-Optic Phase Modulator (EOM)
The EOM is a device that is complementary to the PS as it performs phase shifts of individual modes in the time
domain, and therefore its operator can be expressed as
ˆ
UEOM =
M−1
O
k=0
exp(iφkˆ
Ntk),(9)
where ˆ
Ntkis the number operator of the tktime-bin mode: ˆ
Ntk=Pnn|ntkihntk|. Within our restriction to the
1-photon subspace, we can write the action of the EOM over time modes |tkias a diagonal matrix Ein the {|tki}
basis as
E|tki=eiφk|tki.(10)
Contrary to the PS case, the set of angles {φk}for the EOM cannot be chosen freely. It is determined by the RF
driving. For instance, in the case of a monochromatic (or single tone) RF driving, the phase φkapplied to the optical
wave can be written as a sinusoidal function of time with frequency Ω
φk=µsin(Ωtk+θ) + φc,(11)
with µ=πVm
Vπthe modulation index proportional to the modulation amplitude Vmand Vπthe half-wave voltage of
the EOM, θ∈[0, π] a constant angle and φc∈[0, π] a constant phase shift applied to all time bins. In practice, µis
limited by the power of the RF source that drives the EOM (typically µis of order 1), and by the EOM characteristics.
The maximum frequency Ω is determined by the frequency response of the EOM.
We can describe the action of the EOM in the frequency basis [16,17] (see Fig. 1) through the Fourier transform
˜
E=F EF †and check that the EOM creates left and right side bands (spaced by Ω) with amplitude decreasing
according to Bessel functions. If the mode spacing δω is set equal to the RF driving single-tone Ω, we have
˜
E|ωji=eiφc
k=dµe+1
X
k=−dµe−1
(eiθ)kJk(µ)|ωj+ki.(12)
The sum can be extended to ±∞ as the additional terms are weighed by vanishing Bessel functions [18].
The EOM is therefore a scatterer in the frequency basis.
D. Orders of magnitude
The PS and the EOM act on conjugate variables (frequency and time), which are related by a Fourier transform,
which results in relationships between bandwidths and resolutions. We denominate ∆ωand δω respectively the total
frequency bandwidth that the PS can operate on and its frequency resolution. On the other hand, if the EOM is
driven at a RF frequency Ω, it can generate sidebands at intervals Ω. This naturally defines a number Mω= ∆ω/Ω
frequency modes as long as δω .Ω, so that the PS can distinguish and address them individually. It is important to
match the number of frequency modes and time modes to make sure that the two devices are acting on two conjugate
bases of the same Hilbert space.
5
FIG. 2: Action of the Pulse-shaper (a.) and Electro Optic Phase Modulator (b.) in the time basis.
In the time domain, the resolution δt will be limited by the characteristic times of the detectors. The period of the
RF frequency driving the EOM is T= 2π/Ω, which in principle defines Mt=T/δt = ∆ω/Ω = Mωtime modes.
A realistic PS ( Finisar WaveShaper 4000A) has a total bandwidth ∆ω= 5.36 THz, and the RF voltage signal that
drives the EOM can be set to be a sine wave with a frequency in the order of 10 GHz (which is close to the frequency
resolution of the PS). From these figures we derived a number of time and frequency modes M=T/δt = ∆ω/δω ≈536
and in our calculations, we chose M= 27= 128 modes.
III. PROBLEM DEFINITION: QUBIT GATE SYNTHESIS WITH EOM AND PS
A. Objective
Our aim is to synthesize single-qubit quantum gates using some combination of the components described in the
previous section namely pulse-shapers (PS) and electro-optic modulators (EOM). A single qubit gate is by definition a
unitary transformation over a 2-dimensional Hilbert space, and is hence represented by a 2×2 unitary matrix whereas
the components can be described by unitaries ( P,˜
P,Eand ˜
E) that can operate over a much larger, M-dimensional
Hilbert space. Describing how to perform single qubit gate synthesis in this context therefore implies to specify several
types of information
•The configuration, i.e. how many components are combined into the unitary. After justifying that the minimum
number of components is 3, we focus in this article on two main configurations, [EPE] and [PEP].
•The values of the different components free parameters, (phases of the form ϕ0, . . . , ϕM−1for the pulse-shaper,
and parameters µ, θ, φcfor an electro-optic modulator driven with a single frequency modulation at Ω).
•The choice of encoding, i.e. a choice of two orthonormal M-dimensional vectors, {|e1i,|e2i}, that constitute the
encoding basis and are identified with the logical qubit basis {|0Li,|1Li}.
As explained in the previous section, we will use both frequency basis {|ωii,|ωji} and time basis {|tii,|tji} as
defined in equations (3) and (4).
B. Performance metrics
For a given configuration (e.g. combination of PS and EOMs), we can define a global operator ˆ
V(Φ) and the
corresponding M×M unitary matrix V(Φ), product of nPS and n0EOMs, depending on many parameters that we
will globally denote as Φ ≡(ϕ1,0, . . . , ϕn,M−1, µ1,0, θ1,0, φc1,0. . . µn0,M−1, θn0,M−1, φcn0,M−1).
The choice of the encoding {|e1i,|e2i} ↔ {|0Li,|1Li} then directly induces a reduced 2×2 matrix W(Φ) defined as
W(Φ) = h0L|ˆ
V(Φ) |0Li h0L|ˆ
V(Φ) |1Li
h1L|ˆ
V(Φ) |0Li h1L|ˆ
V(Φ) |1Li.(13)
摘要：

ParallelizableSynthesisofArbitrarySingle-QubitGateswithLinearOpticsandTime-FrequencyEncodingAntoineHenry,1,2RaviRaghunathan,1GuillaumeRicard,1BaptisteLefaucher,1FilippoMiatto,1NadiaBelabas,2IsabelleZaquine,1andRomainAlleaume11TelecomParis-LTCI,InstitutPolytechniquedeParis,19PlaceMargueritePerey,9...

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