Ultra-large-scale continuous-variable cluster states multiplexed in the time domain

TL;DR: In this paper, a continuous-variable cluster state containing more than 10,000 entangled modes is deterministically generated and fully characterized, and an efficient scheme for measurement-based quantum computation on this cluster state is presented.

Abstract: A continuous-variable cluster state containing more than 10,000 entangled modes is deterministically generated and fully characterized. The developed time-domain multiplexing method allows each quantum mode to be manipulated by the same optical components at different times. An efficient scheme for measurement-based quantum computation on this cluster state is presented.

Summary

Introduction

• While such methods are potentially scalable, current experimental results are limited to generating entangled states of just a few modes each21–23.
• The XEPR states are generated by entangling together sequentially propagating EPR states contained within two beams.
• Linear combinations of the quadrature amplitudes at neighbouring times exhibit quantum correlations as in equation (2) and are shown in Fig. 3(e,f).

Methods

• Homodyne detection is employed to measure the quadrature amplitudes of each wave-packet.
• The signals of the homodyne detectors are integrated with the non-overlapping temporallylocalised mode functions of the wave-packets.

S1 Experimental Setup

• Figure S5 shows the schematic of the experimental setup.
• The rest of the fundamental beam is further split and distributed for controlling the sub-threshold optical parametric oscillators (OPOs), the interferometers, the homodyne detectors, and so on.
• The signals from the homodyne detectors in the time-domain are stored by an oscilloscope (DPO 7054, Tektronix).
• For each quadrature measurement of each wavepacket the authors measure 3, 000 frames in order to gather enough statistics to calculate variances.

S2.1 Derivation of Extended EPR States

• An equivalent linear optics network to their experimental setup is represented in Fig. S7.
• In this section the authors derive the expressions of the extended EPR state by following this circuit with both Beam splitter Fiber delay line Optical parametric oscillator Homodyne detection Squeezed mode EPR mode Graph mode Entanglement strengths +1 - 1/2 +1/2 Fig. S6. Legend for animation of experimental setup.
• In the Schrödinger picture, the authors assume the ideal case where the resource squeezing levels are infinite.
• On the other hand the authors can calculate experimentally realistic expressions in the Heisenberg picture.

S2.1.1 Schrödinger Picture in the Ideal Case

• Here the authors utilise infinite squeezing for simplicity.
• As per the following calculations with Schrödinger evolution, the output state is a simultaneous eigenstate of nullifiers.
• First, there are position and momentum eigenstates with zero eigenvalue in each temporal location k.
• Each row in Fig. S7 shows the spatial mode index which the temporal-mode method would correspond to.
• The equivalent quantum circuit to the temporal-mode method setup.

S2.1.2 Heisenberg Evolution with Finite Squeezing

• In the Heisenberg evolution, the variances of nullifiers in the case of finite resource squeezing levels can be calculated.
• S2.2 Inseparability Criteria for Extended EPR States First, the authors consider the combinations of four nodes {Ak, Bk, Ak+1, Bk+1} distributed into the two subsystems.
• Then, the authors apply the same discussion for all temporal indices k.

S2.3 Graph Correspondence

• The authors discuss the intuitive representation of the extended EPR state in terms of the graphical calculus for Gaussian pure states28.
• In that proposal, it was shown that such a state is locally equivalent (up to phase shifts on half the modes) to a CV cluster state, which is a universal resource for measurement-based quantum computing with continuous variables8,27.
• The full graph28 for the extended EPR state.
• The blue edges have positive- (negative-)real weight ±1 2 tanh 2r, and the green self-loops have positive-imaginary weight i cosh 2r.

S2.4 Equivalence to Sequential Teleportation-based Quantum Computation Circuit

• In reference 25, Menicucci proposed that by erasing half of the state (one rail), the cluster states can be used as resources for measurement-based quantum computation (MBQC).
• The authors devise a much more efficient method of using this resource state for quantum computation than the method originally proposed in ref. 25, in terms of its use of the available squeezing resources.
• Specifically, arbitrary Gaussian operation may be implemented by only 4 measurements, which is more efficient than the 8 measurements necessary with the original method30.
• Furthermore, the authors show that non-Gaussian operations may be performed on the extended EPR state by introducing non-Gaussian measurements, leading to one-mode universal MBQC.

S2.4.1 Gaussian Operation

• First of all, let us consider the quantum teleportation-based circuit shown in Fig. S11.
• This shows that the extended EPR state can be used as a resource for MBQC.

S2.4.2 Non-Gaussian Operation

• Non-Gaussian operations may also be implemented by using the teleportation-based circuit shown in Fig. S14.
• Note that since it can be accomplished by only using displacement feedforwards, input coupling beam-splitters can also be exchanged.
• Therefore, by using the extended EPR state, non-Gaussian operations can be performed sequentially, resulting in a resource for universal one-mode universal MBQC.

S3 Data Analysis

• S3.1 Influence of Experimental Losses Experimental imperfections lead to degraded resource squeezing levels.
• In particular, the unbalanced losses between the optical fiber and free space channels cause the degradation of nullifier variances.

Figures

Figure 1 | Schematic of the experimental setup and ultra-large-scale XEPR state. a, Generation of ultra-large-scale cluster-state entanglement. Squeezed temporal modes are mixed on a beam-splitter to create EPR states, which are then staggered by a fiber delay line. The staggered EPR-state wave-packets are then mixed to generate a continuous graph structure, where each wave-packet is entangled to neighbouring wave-packets. Nodes (coloured spheres) and links between them represent optical wave-packets and entanglement, respectively. Independent quantum states exist in each temporally localised wave-packet with time duration T . OPO, optical parametric oscillator; 50:50 BS, balanced beam-splitter; HD, homodyne detector; LO, local oscillator. b, The extended EPR state, equivalent to a CV cluster state up to local phase shifts (redefinition of quadrature operators). See supplementary section ‘S2’ for full details and graph representation.

Figure 4 | Quantum correlations of the XEPR state for the first 30, 000 wave-packets. a,b, (i) Measured nullifier variances for the first 15, 000 temporal mode positions (30, 000 wavepackets in both rails) are shown for the x̂ quadrature in blue (a) and for the p̂ quadrature in red (b), as in equation (3). (ii) measured variances for vacuum inputs, used as a reference for noise power. (iii) −3.0 dB lines indicate the bounds of inseparability; data points below this line demonstrate entanglement.

Figure 3 | Measured quantum correlations of the first 50 wave-packets. a–d, Measured quadrature amplitudes of x̂Ak , p̂Ak , x̂Bk and p̂Bk of temporally localised wavepackets at temporal node positions k. They are randomly distributed around zero. e,f, Additions and time-shifted subtractions of certain quadratures show quantum correlations displayed by the near perfect overlaps.

Figure 2 | XEPR states for sequential quantum teleportation. a, Circuit of standard sequential quantum teleportation. b, Circuit of sequential quantum teleportation through the XEPR state. 50:50 BS, balanced beam-splitter; |ψ〉, arbitrary and unknown quantum state as the input of quantum teleporter; X̂ or Ẑ, phase-space displacement operation based on measurement outcomes.

Full text

Please do not remove this page
Ultra-large-scale continuous-variable cluster
states multiplexed in the time domain
Yokoyama, Shota; Ukai, Ryuji; Armstrong, Seiji; Sornphiphatphong, Chanond; Kaji, Toshiyuki; Suzuki,
Shigenari; Yoshikawa, Jun-ichi
https://researchrepository.rmit.edu.au/discovery/delivery/61RMIT_INST:ResearchRepository/12247914280001341?l#13248404440001341
Yokoyama, Ukai, R., Armstrong, S., Sornphiphatphong, C., Kaji, T., Suzuki, S., Yoshikawa, J., Menicucci, N., &
Furusawa, A. (2013). Ultra-large-scale continuous-variable cluster states multiplexed in the time domain.
Nature Photonics, 7(12), 982–986. https://doi.org/10.1038/nphoton.2013.287
Published Version: https://doi.org/10.1038/nphoton.2013.287
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https://researchbank.rmit.edu.au/view/rmit:34276
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ccepted Manuscript
2013 Macmillan Publishers Limited. All rights reserved.
https://dx.doi.org/10.1038/nphoton.2013.287
Yokoyama, S, Ukai, R, Armstrong, S, Sornphiphatphong, C, Kaji, T, Suzuki, S, Yoshikawa, J,
Menicucci, N and Furusawa, A 2013, 'Ultra-large-scale continuous-variable cluster states multiplexed
in the time domain', Nature Photonics, vol. 7, pp. 982-986.

Ultra-Large-Scale Continuous-Variable Cluster States
Multiplexed in the Time Domain
Shota Yokoyama
1
, Ryuji Ukai
1
, Seiji C. Armstrong
1,2
,
Chanond Sornphiphatphong
1
, Toshiyuki Kaji
1
, Shigenari Suzuki
1
,
Jun-ichi Yoshikawa
1
, Hidehiro Yonezawa
1
,
Nicolas C. Menicucci
3
, Akira Furusawa
1∗
1
Department of Applied Physics, School of Engineering, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
2
Centre for Quantum Computation and Communication Technology,
Department of Quantum Science, The Australian National University,
Canberra, ACT 0200, Australia
3
School of Physics, The University of Sydney, NSW, 2006, Australia
∗
To whom correspondence should be addressed; E-mail: akiraf@ap.t.u-tokyo.ac.jp
1

Quantum computers promise ultrafast performance of certain tasks
1
. Experimentally
appealing, measurement-based quantum computation (MBQC)
2
requires an entangled
resource called a cluster state
3
, with long computations requiring large cluster states. Pre-
viously, the largest cluster state consisted of 8 photonic qubits
4
or light modes
5
, while the
largest multipartite entangled state of any sort involved 14 trapped ions
6
. These imple-
mentations involve quantum entities separated in space, and in general, each experimental
apparatus is used only once. Here, we circumvent this inherent inefficiency by multiplex-
ing light modes in the time domain. We deterministically generate and fully characterise a
continuous-variable cluster state
7,8
containing more than 10,000 entangled modes. This is,
by 3 orders of magnitude, the largest entangled state ever created to date. The entangled
modes are individually addressable wavepackets of light in two beams. Furthermore, we
present an efficient scheme for MBQC on this cluster state based on sequential applica-
tions of quantum teleportation.
Originally formulated as a demonstration as to why quantum mechanics must be incomplete
in the famous 1935 Einstein-Podolsky-Rosen (EPR) paradox
9
, entanglement is now recognized
as a signature feature of quantum physics
10
, and it plays a central role in various quantum infor-
mation processing (QIP) protocols
1,11
. For example, the bipartite entangled state known as an
EPR state
9
is a resource for quantum teleportation (QT), whereby a quantum state is transferred
from one location to another without physical transfer of the quantum information
12–14
.
Measurement-based quantum computation (MBQC)
2,7,8,15–18
, which is based on the QT of
information and logic gates, requires the special class of multipartite entangled resource states
known as cluster states
3
. The number of entangled quantum entities and their entanglement
structure (represented by a graph) determines the resource space available for computation.
Ultra-large-scale QIP (which could be based on MBQC) will require ultra-large-scale entangled
2

states
2,7,8
.
In the vast majority of optical experiments, quantum modes are distinguished from each
other by their spatial location. This leads to an inherent lack of scalability as each additional
entangled party requires an increase in laboratory equipment and dramatically increases the
complexity of the optical network
19,20
. Further, due to the probabilistic nature of photon pair
generation, demonstrations involving the postselection of photonic qubits
4,15,16
suffer from dra-
matically reduced event success rates with each additional qubit.
One method to overcome this problem of scalability is to deterministically encode the
modes within one beam. Entanglement between quadrature-phase amplitudes in continuous-
wave laser beams has been deterministically created and exploited in QIP
5,13,14,17–19,21–23
, even
though the quantum correlations are finite. Previous attempts to deterministically create cluster
states within one beam have exploited the spatial
21
or spectral
22–24
orthogonality of quantum
modes. While such methods are potentially scalable, current experimental results are limited to
generating entangled states of just a few modes each
21–23
. A novel method proposed in ref. 25
lets quantum modes propagate within the same beam — distinguished and ultimately made
orthogonal by their separation in time. The time-domain multiplexing approach allows each
additional quantum mode to be manipulated by the same optical components at different times,
which is a powerful concept, as found in atomic ensemble quantum memories
26
.
Here, we demonstrate the deterministic generation of ultra-large-scale entangled states con-
sisting of more than 10, 000 entangled wave-packets of light, multiplexed in the time domain.
The generated states, which we call extended EPR (XEPR) states, are equivalent up to local
phase shifts to topologically one-dimensional continuous-variable (CV) cluster states
25
and are
therefore a universal resource for single-mode MBQC with continuous-variables
8
. Fully univer-
sal multimode MBQC is achievable simply by combining two XEPR states with differing time
delays on two additional beam-splitters
25
. Note that in our time-domain multiplexed demonstra-
3

Frequently asked questions

Q1. What is the phase modulation technique used for locking the optical cavities?

The phase modulation by the EOM adds 16.5 MHz sideband components which are utilised for locking all the optical cavities via the Pound-Drever-Hall locking technique. 

Q2. What is the optical pass length of the fiber?

Since small temperature changes around the fiber cause drastic changes of the optical pass length resulting in the instability of phase locking, the fiber is placed inside a box consisting of heat insulating material and vibration-proofing materials. 

Q3. How many frames are used to measure the variances of the wavepackets?

For each quadrature measurement of each wavepacket the authors measure 3, 000 frames in order to gather enough statistics to calculate variances. 

Q4. What is the weight of the green self-loops?

The blue (yellow) edges have positive- (negative-)real weight ±1 2 tanh 2r, and the green self-loops have positive-imaginary weight i cosh 2r. 

Q5. What is the special case of the N -mode ground state?

The special case of the N -mode ground state (Zground = iI) is easy to verify by noting that the vector of nullifiers in that case is just the vector of annihilation operators. 

Q6. What is the simplest way to define a CV cluster state graph?

In the large-squeezing limit, t = tanh 2r → 1, and ε = sech 2r → 0, which allows us to define an unphysical, ideal CV cluster-state graph G to which ZC is a physical approximation: 

Q7. How can the authors transform the state of the CV cluster?

This state can be transformed, by −π 2 phase shifts on both modes of all odd (or all even) time indices, into the following CV cluster state ZC (ref. 25):i!–t/2t/2i!–t/2–t/2t/2t/2ZC 

Q8. How can one account for the phase shifts in the extended EPR state?

one can account for them entirely just by updating the measurement-based protocol to be implemented (i.e., redefine quadratures x̂ → p̂ and p̂ → −x̂ on the appropriate modes)25. 

Q9. What is the difference between the extended EPR state and the corresponding CV cluster state?

Since measurement-based quantum computation requires the ability to do homodyne detection of any (rotated) quadrature, plus photon counting27, the phase shifts required to transform the generated state (the extended EPR state) into a CV cluster state do not need to be physically performed on the state after generation. 

Q10. What is the smallest nullifier in the MBQC?

Substituting Z → ZC [Eq. (S33)] and noting that Z−1C = −i(sech 2r)I + (tanh 2r)G, the exact nullifiers are(−iεx̂ + p̂− tGx̂) |ψZC 〉 = 0,(iεp̂ + x̂− tGp̂) |ψZC 〉 = 0, (S38)which, in the large-squeezing limit, reduce to the following approximate nullifiers:(p̂−Gx̂) |ψZC 〉 r→∞−−−→ 0, (x̂−Gp̂) |ψZC 〉 r→∞−−−→ 

Q11. What is the method for using the extended EPR state?

The authors devise a much more efficient method of using this resource state for quantum computation than the method originally proposed in ref. 25, in terms of its use of the available squeezing resources.