Continuous-variable quantum computing: scalable designs and fault - - PowerPoint PPT Presentation

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Continuous-variable quantum computing: scalable designs and fault - - PowerPoint PPT Presentation

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Continuous-variable quantum computing: scalable designs and fault tolerance Nicolas C Menicucci RIT Jul 2020 (Comic sans forever!) My group https://www.qurmit.org/ 2 ARC Centre of Excellence https://www.cqc2t.org/ 3 What is

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Continuous-variable quantum computing:
 scalable designs and fault tolerance

Nicolas C Menicucci

(Comic sans forever!)

RIT Jul 2020

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My group

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https://www.qurmit.org/

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ARC Centre of Excellence

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https://www.cqc2t.org/

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What is Computation?

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What is Computation?

abstract:

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What is Computation?

input (information)

abstract:

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What is Computation?

input (information)

  • utput

(information)

abstract:

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What is Computation?

input (information)

  • utput

(information)

abstract: physical:

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What is Computation?

input (information)

  • utput

(information)

abstract: physical:

initial state of a physical system

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What is Computation?

input (information)

  • utput

(information)

abstract: physical:

initial state of a physical system final state of a physical system

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physical process

What is Computation?

input (information)

  • utput

(information)

abstract: physical:

initial state of a physical system final state of a physical system

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physical process

What is Computation?

input (information)

  • utput

(information)

abstract: physical:

initial state of a physical system final state of a physical system computation

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Qubits and CVs

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Qubits and CVs

quantum
 computation input (information)

  • utput

(information)

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Qubits and CVs

quantum
 computation input (information)

  • utput

(information)

Qubits

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Qubits and CVs

quantum
 computation input (information)

  • utput

(information)

Qubits Unitary evolution

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Qubits and CVs

quantum
 computation input (information)

  • utput

(information)

CVs Qubits Unitary evolution

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Qubits and CVs

quantum
 computation input (information)

  • utput

(information)

CVs Unitary evolution Qubits Unitary evolution

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Cluster states

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Teleportation “Lite”

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Teleportation “Lite”

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Teleportation “Lite”

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Teleportation “Lite”

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Teleportation “Lite”

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Teleportation “Lite”

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Teleportation Network

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Cluster State

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Cluster State

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Cluster State

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Cluster State

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Cluster State

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Cluster state

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Cluster state

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Cluster state

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Cluster state

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Why bother with CVs?

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CVs: Advantages

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CVs: Advantages

■ Practical

  • deterministic entanglement
  • huge scaling potential
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CVs: Advantages

■ Practical

  • deterministic entanglement
  • huge scaling potential

■ Fundamental

  • avoid premature optimisation


(i.e., why should we restrict to photonic qubits?)

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CVs: Advantages

■ Practical

  • deterministic entanglement
  • huge scaling potential

■ Fundamental

  • avoid premature optimisation


(i.e., why should we restrict to photonic qubits?)

■ Both together

  • more options for practical tasks (e.g., quantum

cryptography, cluster states)

  • "hybrid" schemes: CV technology helps to manipulate

photonic quantum states

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CVs: Disadvantages

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CVs: Disadvantages

■ Practical

  • intrinsic noise due to finite squeezing (more later)
  • eventually need to discretise for error correction

(more later)

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CVs: Disadvantages

■ Practical

  • intrinsic noise due to finite squeezing (more later)
  • eventually need to discretise for error correction

(more later)

■ Fundamental

  • more questions to answer (e.g., what discretisation?)
  • must incorporate effects of noise from day one

(complicated, easy to end up writing a crap paper)

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CVs: Disadvantages

■ Practical

  • intrinsic noise due to finite squeezing (more later)
  • eventually need to discretise for error correction

(more later)

■ Fundamental

  • more questions to answer (e.g., what discretisation?)
  • must incorporate effects of noise from day one

(complicated, easy to end up writing a crap paper)

■ Both together

  • must do extra work to employ existing algorithms
  • smaller literature, fewer optimised experimental platforms
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CV cluster states

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CV cluster states

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CV cluster states

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Optical implementation

■ Continuous quantum variables

  • Computational basis: eigenstates of q = (a + a†)/√2
  • Conjugate basis:

eigenstates of p = –i(a – a†)/√2

■ Advantages of CV (over qubit) cluster states

  • Deterministic generation
  • Scalable to huge sizes

■ Problem: ideal CV cluster states are infinitely

squeezed (infinite energy)

  • Finite squeezing → additive Gaussian noise
  • Fault tolerance possible with encoded qubits

(GKP)

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Computation with ideal states

■ Single-mode projective measurements are

sufficient for universal QC

■ Homodyne detection (quadrature

measurements) enables all Gaussian unitaries

  • Relatively easy to do experimentally
  • Very low noise

■ Photon counting enables the rest

  • Less efficient, but technology rapidly improving

■ Still have to handle intrinsic noise…

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Fault tolerance

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Encoded qubits

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Encoded qubits

q

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Noise process

Information flow

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Noise process

Information flow

1 2 3 4 5 6 7 8 9

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Projecting to qubit-level errors

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Projecting to qubit-level errors

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Projecting to qubit-level errors

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Projecting to qubit-level errors

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Projecting to qubit-level errors

Recovery
 (no error)

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Projecting to qubit-level errors

Qubit-level
 error

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Noisy qubit-level Clifford gates

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Noisy qubit-level Clifford gates

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Noisy qubit-level Clifford gates

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Noisy qubit-level Clifford gates

Noisy
 1-qubit Clifford
 gate

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Noisy qubit-level Clifford gates

Noisy
 1-qubit Clifford
 gate Noisy
 2-qubit Clifford
 gate

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Fault tolerance

■ Homodyne detection implements (faulty) qubit

gates

■ Use qubit-level quantum error correction to

reduce errors (well established)

■ Fault tolerance


(initial error < threshold) →
 (arbitrarily low error in final computation)

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Fault tolerance

High squeezing ⇒ low Gaussian noise ⇒ low rate of Pauli errors

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Fault tolerance

High squeezing ⇒ low Gaussian noise ⇒ low rate of Pauli errors How
 high?

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Fault tolerance

High squeezing ⇒ low Gaussian noise ⇒ low rate of Pauli errors How
 high?

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Fault tolerance

Fault-tolerant squeezing threshold ≤ 20.5 dB

High squeezing ⇒ low Gaussian noise ⇒ low rate of Pauli errors How
 high?

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Resources sufficient for FT QC

■ CV cluster state with sufficient squeezing:

"railroad tracks"

■ GKP qubits with sufficient squeezing:


"train cars" carrying the discrete quantum information

■ Homodyne detection = Gaussian unitaries:


"switches" to guide the info & measurement

■ Non-Clifford resource: photon counting, cubic-

phase gate, cubic phase state

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Resources sufficient for FT QC

■ CV cluster state with sufficient squeezing:

"railroad tracks"

■ GKP qubits with sufficient squeezing:


"train cars" carrying the discrete quantum information

■ Homodyne detection = Gaussian unitaries:


"switches" to guide the info & measurement

■ Non-Clifford resource: photon counting, cubic-

phase gate, cubic phase state

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Resources sufficient for FT QC

■ CV cluster state with sufficient squeezing:

"railroad tracks"

■ GKP qubits with sufficient squeezing:


"train cars" carrying the discrete quantum information

■ Homodyne detection = Gaussian unitaries:


"switches" to guide the info & measurement

■ Non-Clifford resource: photon counting, cubic-

phase gate, cubic phase state

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Resources sufficient for FT QC

■ CV cluster state with sufficient squeezing:

"railroad tracks"

■ GKP qubits with sufficient squeezing:


"train cars" carrying the discrete quantum information

■ Homodyne detection = Gaussian unitaries:


"switches" to guide the info & measurement

■ Vacuum state! (or heterodyne detection)

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Experimental GKP states

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Experimental GKP states

|0L⟩ |1L⟩

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Experimental GKP states

arXiv:1907.12487

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Experimental GKP states

arXiv:1907.12487

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Making CV cluster states

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Linear optics

■ Inline squeezing (CZ gate) can be replaced

with offline squeezing + interferometer*

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Linear optics

■ Inline squeezing (CZ gate) can be replaced

with offline squeezing + interferometer*

S S S S S

* P. van Loock, C. Weedbrook, M. Gu, PRA 76, 032321 (2007)

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How can we make
 scalable resource states?

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Macronode-based cluster states

Temporal or frequency modes

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Frequency-mode cluster states

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Frequency-mode cluster states

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Frequency-mode cluster states

60-mode linear cluster state

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Frequency-mode cluster state (wire)

Pump Laser*1 Pump Laser*2 Pump*1 Pump*2

OPO pump cluster state frequency- sensitive measurements

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Homodyne detectors Pumps OPO Measurement selection

Frequency-mode cluster state (wire)

OPO pump cluster state frequency- sensitive measurements

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Frequency-mode cluster state (wire)

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Frequency-mode cluster state (wire)

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Frequency-mode cluster state (wire)

equivalent

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Frequency-mode cluster state (wire)

equivalent 60 modes
 addressable

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Frequency-mode cluster state (wire)

equivalent after
 polarization
 rotation (verification) 60 modes
 addressable

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Temporal-mode cluster states

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Temporal-mode cluster states

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Temporal-mode cluster states

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Temporal-mode cluster states

10,000-mode linear cluster state

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Temporal-mode cluster states

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Temporal-mode cluster states

Animation by Seiji Armstrong (available online at https://youtu.be/gor29QIP9Ls)

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Temporal-mode cluster states

Animation by Seiji Armstrong (available online at https://youtu.be/gor29QIP9Ls)

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Temporal-mode cluster states

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Temporal-mode cluster states

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Temporal-mode cluster states

1-million-mode linear cluster state!

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Temporal-mode cluster states

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Temporal-mode cluster states

24,800 total modes
 5 x 1240 macronodes
 (4 modes each)

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Temporal-mode cluster states

30,000 total modes
 12 x 1250 macronodes
 (2 modes each)

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Other proposed approaches

■ Frequency-temporal modes

  • like musical tones: one frequency for some

period of time

  • frequency adds an extra lattice dimension

■ Frequency-spatial modes

  • frequency-encoded linear states in different

beams woven together

■ Three-dimensional lattice topologies

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Conclusion

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Conclusion

■ CV cluster states

  • Enable measurement-based quantum computation using

continuous variables

  • Fault tolerance is possible

■ GKP qubits

  • Enable fault tolerance with CV cluster states
  • All-Gaussian gate set (only known bosonic code with this feature!)
  • Achieved in trapped ions and circuit QED

■ Macronode-based methods are scalable

  • 1D CV cluster state (wire): frequency and temporal modes achieved
  • 2D CV cluster state (universal): temporal modes achieved
  • 3D and higher-dimensional lattices possible
  • Millions of modes achieved
  • Need to improve squeezing (~4.5 dB, need >10 dB)
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58 AIP, Dec 2014