Recursive quantum algorithms for integrated linear optics Gelo - - PowerPoint PPT Presentation

Recursive quantum algorithms for integrated linear optics

Gelo Noel Tabia

28th CS Theory Days | 2-4 October 2015

Motivation

• Photons make excellent qubits

(robust against decoherence).

• Universal, scalable quantum

computing is possible with linear

• ptics (KLM, cluster states)
• Goal: recipe for translating quantum

algorithms into a practical linear

• ptical scheme

Outline

• Quantum circuits
• Linear optics
• Photonic integrated circuit
• Quantum Fourier transform
• Grover’s algorithm

Qubit

• 2-level quantum system
• Photon polarization, dual-rail qubit

𝜔 = 𝛽 0 + 𝛾 1 = 𝛽 𝛾 , 𝛽 2+ 𝛾 2 = 1

horizontal vertical mode 0 mode 1

1 1

Unitary gates

• Norm-preserving linear operation
• If 𝑉 is unitary then

𝑉 𝛽 𝛾 ↦ 𝛿 𝜀 , 𝛿 2 + 𝜀 2 = 1

𝑉𝑉† = 𝑉†𝑉 = 𝐽

Qubit gates

• Single-qubit and 2-qubit gates

𝐼 = 1 2 1 1 1 −1 𝑈 = 1 𝑓𝑗𝜌/4

Hadamard gate 𝜌/8-phase gate

CNOT 𝑦 𝑧 = 𝑦 |𝑧 ⊕ 𝑦〉

Controlled-NOT gate

CNOT = 1 1 1 1

Measurement

• Orthogonal projectors onto subspaces
• If 𝑄

𝑘 are measurement projectors

Π0 = 0 0 = 1 Π1 = 1 1 = 0 1 Pr 𝑘 = 𝜔 Π𝑘|𝜔〉 𝜔 = 𝜔 †

𝑘

𝑄

𝑘 = 𝐽

Quantum circuit

• Prepare input qubits
• Apply sequence of quantum gates
• Measure output qubits

Π𝑗

|𝜔𝑜〉 |0〉

𝑉𝜔𝑜

Quantum algorithm

• Quantum circuit family
• Consistent
• Uniform:

𝐷𝑜+𝑛 𝜔𝑜 ⊗ 0 ⊗𝑛 = Cn 𝜔𝑜 ⊗ 0 ⊗𝑛 𝐷𝑜: 𝐷𝑜 is a quantum circuit for 𝑜 qubits ∃ efficient algorithm for building 𝐷𝑜 given 𝑜

Qudits

• 𝑒-level quantum system

𝜔 = 𝛽 𝛾 𝛿

1 2

𝐺3 = 1 3 1 1 1 1 𝜕 𝜕2 1 𝜕2 𝜕 Π0 = 0 0 + 1 〈1| Π1 = 2 2

e.g. qutrit for 𝑒 = 3

Linear optics (LO)

• Photons manipulated by a network of

beam splitters, phase shifters, mirrors

e.g., Mach-Zehnder interferometer

Phase shifter

• Phase 𝜄

𝑄𝜄 = 1 𝑓𝑗𝜄

𝜄 Phase 𝑓𝑗𝜄 applied to amplitude in mode 2 𝜄 = 2𝜌𝑀(𝑜 − 1)/𝜇

Beam splitter

• Reflectivity 𝜗

𝜗

𝐶𝜗 = 𝜗 1 − 𝜗 1 − 𝜗 − 𝜗

Photon in mode 1 stays there with probability 𝜗 𝜗 1 − 𝜗

Photonic integrated circuit

• Waveguide-based linear optics

Universal LO processor

• J. Carolan et al. Science, 349, (2015) 711-716

Experimental tests

• Hadamard matrices
• Boson sampling

LO unitary gates

• Any 𝑒-dimensional unitary can be

realized with a LO network on 𝑒 modes using 𝑒2 − 1 elements

• For arbitrary 𝑉 ∈ 𝑇𝑉 𝑒 , we need to

specify 𝑒2 − 1 real parameters.

• Reck, et al. (1994):

Our result

• We describe recursive LO circuits for

quantum Fourier transform and Grover inversion

• We build the circuit for 2𝑒 modes

using a pair of circuits for the same

• peration on 𝑒 modes.
• Formally, we decompose unitaries

into a product of block-diagonal matrices with adjacent 2 × 2 blocks

Quantum Fourier transform

• Discrete Fourier transform on

quantum states

𝐺

4 = 1

4 1 1 1 𝑗 1 1 −1 −𝑗 1 −1 1 −𝑗 1 −1 −1 𝑗

Recursive QFT circuit

• QFT for 𝑒 = 8 given the circuit for 𝐺

4

Swap operator, same as 𝐶0

LO for QFT

• Let Σ denote the permutation
• Let Σ−1 be the inverse of Σ.
• Let 𝑄𝜄 𝑘 be a phase shift 𝜄 on mode 𝑘.
• Let 𝐶𝜗(𝑗, 𝑘) denote a beam splitter with

reflectivity 𝜗 on modes 𝑗, 𝑘. 1,2, … , 2𝑒 ↦ (1, 𝑒 + 1,2, 𝑒 + 2, … . , 𝑒, 2𝑒)

Constructing 𝐺

2𝑒 given 𝐺 𝑒

1) LO circuit for Σ−1 2) Apply 𝐺

𝑒 on modes 1 to 𝑒 and 𝐺 𝑒 on modes 𝑒 + 1

to 2𝑒 3) Use the following phase shifters: 4) LO circuit for Σ. 5) Use the following beam splitters: 6) LO circuit for Σ−1 𝑄𝜌

𝑒(𝑒 + 2),…, 𝑄𝑙𝜌 𝑒

(𝑒 + 𝑙 + 1),…, 𝑄(𝑒−1)𝜌

𝑒

(2𝑒) 𝐶1

2 1,2 , 𝐶1 2(3,4),…, 𝐶1 2(2𝑒 − 1,2𝑒)

Fourier matrix factorization

• First discovered by Gauss, this is the

basis for fast Fourier transform 𝐺2𝑒 = 𝐽 𝐸 𝐸 𝐽 𝐺𝑒 𝐺𝑒 Σ−1 𝐸 = diag(1, 𝜕, … , 𝜕𝑒)

Search problem

• For unsorted database search
• Given black box for 𝑔: 0,1 𝑜 → 0,1
• Find an 𝑦 s.t. 𝑔(𝑦) = 1, if any.
• Classically, Ω(𝑜) queries are needed.
• In the quantum case, 𝑃

𝑜 are sufficient

Grover iterate

• Oracle 𝑉

𝑔: 𝑦 ↦ −1 𝑔(𝑦) 𝑦

• Let 𝑇: 𝑦 ↦ − 𝑦 , 𝑦 ≠ 0

0 ↦ 0

• Define Grover inversion (GI)
• Grover iterate

𝑋 = 𝐼⊗𝑜𝑇𝐼⊗𝑜 = 2 𝜔 𝜔 − 𝐽

𝜔 : = 1 2𝑜

𝑦

|𝑦〉

𝐻 = 𝑋𝑉

𝑔

Grover’s algorithm

𝐻 = 𝑋𝑉

𝑔

• We construct a recursive LO circuit for

Grover inversion 𝑋

𝑒

LO for unitary 𝑊

𝑒

• First, define family of unitaries 𝑊

𝑒:

• LO circuit for 𝑊

4

Recursive 𝑊

𝑒 circuit

• For 𝑒 = 8 given the circuit for 𝑊

4

Constructing 𝑊

2𝑒 given 𝑊 𝑒

1) Apply 𝑊

𝑒 on modes 1 to 𝑒 and 𝑊 𝑒 on modes 𝑒 + 1

to 2𝑒. 2) LO circuit for Σ. 3) Use the following beam splitters: 4) LO circuit for Σ−1. 𝐶1

2 1,2 , 𝐶1 2(3,4),…, 𝐶1 2(2𝑒 − 1,2𝑒)

LO for Grover inversion

• LO circuit for 𝑋

4

𝑋

4 = 1

2 −1 1 1 −1 1 1 1 1 1 1 1 1 −1 1 1 −1

Recursive 𝑋

𝑒 circuit

• 𝑋

8 given the circuit for 𝑋 4 and 𝑊 4

Constructing 𝑋

2𝑒 given 𝑋 𝑒

1) Apply 𝑋

𝑒 on modes 1 to 𝑒 and 𝑋 𝑒 on modes 𝑒 + 1

to 2𝑒. 2) Apply 𝑊

𝑒 on modes 1 to 𝑒 and 𝑊 𝑒 on modes 𝑒 + 1

to 2𝑒. 3) Let Φ be the permutation that exchanges mode 1 and mode 𝑒 + 1. Use the LO circuit for Φ. 4) Apply 𝑊

𝑒 on modes 1 to 𝑒 and 𝑊 𝑒 on modes 𝑒 + 1

to 2𝑒.

𝑋

𝑒 matrix decomposition

• Our scheme implies the factorization

where 𝑋

2 = 0

1 1 0 , 𝑊

2 = 𝐼.

𝑋

2𝑒 = 𝑊 𝑒

𝑊

𝑒 Φ 𝑊 𝑒

𝑊

𝑒

𝑋

𝑒

𝑋

𝑒

𝑊

2𝑒 = 𝐼 ⊗ 𝐽𝑒

𝑊

𝑒

𝑊

𝑒

Simulation

8-item Grover search

LO circuit for 𝑄8

• Prepares an equal superposition state

Simulation results

• Fidelity 𝑔 = 〈theo|expt〉

𝑂 = 107 𝜈 = 0.871 𝜏 = 0.059

Error model

• 4% on BS reflectivities
• 5% absorption loss in PS

Open problems

• Recursive LO circuit for other

interesting unitaries, e.g.,

• Given some unitary U what is the

minimum number of optical elements needed for its LO circuit? 𝑉𝑏: 𝑡 ↦ 𝑡𝑏 (mod 𝑂) , 0 ≤ 𝑡 < 𝑂