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 optics (KLM, cluster states) • Goal: recipe for translating quantum algorithms into a practical linear optical scheme

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

Qubit • 2-level quantum system 𝜔 = 𝛽 0 + 𝛾 1 = 𝛽 𝛽 2 + 𝛾 2 = 1 𝛾 , • Photon polarization, dual-rail qubit 0 0 1 1 vertical horizontal mode 1 mode 0

Unitary gates • Norm-preserving linear operation 𝑉 𝛽 𝛾 ↦ 𝛿 𝛿 2 + 𝜀 2 = 1 𝜀 , • If 𝑉 is unitary then 𝑉𝑉 † = 𝑉 † 𝑉 = 𝐽

Qubit gates • Single-qubit and 2 -qubit gates 𝐼 = 1 1 1 𝑈 = 1 0 𝑓 𝑗𝜌/4 1 −1 0 2 𝜌/8 -phase gate Hadamard gate 1 0 0 0 CNOT 𝑦 𝑧 0 1 0 0 CNOT = 0 0 0 1 = 𝑦 |𝑧 ⊕ 𝑦〉 0 0 1 0 Controlled-NOT gate

Measurement • Orthogonal projectors onto subspaces Π 0 = 0 0 = 1 0 0 0 Pr 𝑘 = 𝜔 Π 𝑘 |𝜔〉 Π 1 = 1 1 = 0 0 𝜔 = 𝜔 † 0 1 • If 𝑄 𝑘 are measurement projectors 𝑄 𝑘 = 𝐽 𝑘

Quantum circuit • Prepare input qubits • Apply sequence of quantum gates • Measure output qubits |𝜔 𝑜 〉 𝑉 𝜔 𝑜 Π 𝑗 |0〉

Quantum algorithm • Quantum circuit family 𝐷 𝑜 : 𝐷 𝑜 is a quantum circuit for 𝑜 qubits • Consistent 𝜔 𝑜 ⊗ 0 ⊗𝑛 = C n 𝜔 𝑜 ⊗ 0 ⊗𝑛 𝐷 𝑜+𝑛 • Uniform: ∃ efficient algorithm for building 𝐷 𝑜 given 𝑜

Qudits • 𝑒 -level quantum system 𝛽 𝛾 𝜔 = 1 1 1 𝐺 3 = 1 𝛿 𝜕 2 1 𝜕 3 𝜕 2 1 𝜕 0 Π 0 = 0 0 + 1 〈1| 1 Π 1 = 2 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 𝑄 𝜄 = 1 0 𝑓 𝑗𝜄 0 𝜄 = 2𝜌𝑀(𝑜 − 1)/𝜇 Phase 𝑓 𝑗𝜄 applied to amplitude in 𝜄 mode 2

Beam splitter • Reflectivity 𝜗 1 − 𝜗 𝜗 𝜗 1 − 𝜗 𝐶 𝜗 = 1 − 𝜗 − 𝜗 𝜗 Photon in mode 1 stays there with probability 𝜗

Photonic integrated circuit • Waveguide-based linear optics

Universal LO processor Experimental tests Hadamard matrices • Boson sampling • J. Carolan et al. Science, 349, (2015) 711-716

LO unitary gates • Any 𝑒 -dimensional unitary can be realized with a LO network on 𝑒 modes using 𝑒 2 − 1 elements • For arbitrary 𝑉 ∈ 𝑇𝑉 𝑒 , we need to specify 𝑒 2 − 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 operation 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 1 1 1 1 4 = 1 1 𝑗 −1 −𝑗 𝐺 1 −1 1 −1 4 1 −𝑗 −1 𝑗

Recursive QFT circuit • QFT for 𝑒 = 8 given the circuit for 𝐺 4 Swap operator, same as 𝐶 0

LO for QFT • Let Σ denote the permutation 1,2, … , 2𝑒 ↦ (1, 𝑒 + 1,2, 𝑒 + 2, … . , 𝑒, 2𝑒) • Let Σ −1 be the inverse of Σ . • Let 𝑄 𝜄 𝑘 be a phase shift 𝜄 on mode 𝑘 . • Let 𝐶 𝜗 (𝑗, 𝑘) denote a beam splitter with reflectivity 𝜗 on modes 𝑗, 𝑘 .

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: 𝑒 (𝑒 + 2) ,…, 𝑄 𝑙𝜌 (𝑒 + 𝑙 + 1) ,…, 𝑄 (𝑒−1)𝜌 𝑄 𝜌 (2𝑒) 𝑒 𝑒 4) LO circuit for Σ. 5) Use the following beam splitters: 2 (3,4) ,…, 𝐶 1 𝐶 1 2 1,2 , 𝐶 1 2 (2𝑒 − 1,2𝑒) 6) LO circuit for Σ −1

Fourier matrix factorization • First discovered by Gauss, this is the basis for fast Fourier transform 𝐺 𝑒 0 𝐽 𝐸 𝐺 𝑒 Σ −1 𝐺 2𝑒 = 0 𝐸 𝐽 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 𝑔 : 𝑦 ↦ −1 𝑔(𝑦) 𝑦 • Oracle 𝑉 • Let 𝑇: 𝑦 ↦ − 𝑦 , 𝑦 ≠ 0 0 ↦ 0 • Define Grover inversion (GI) 𝑋 = 𝐼 ⊗𝑜 𝑇𝐼 ⊗𝑜 = 2 𝜔 𝜔 − 𝐽 1 • Grover iterate 𝜔 : = 2 𝑜 |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: 2 (3,4) ,…, 𝐶 1 𝐶 1 2 1,2 , 𝐶 1 2 (2𝑒 − 1,2𝑒) 4) LO circuit for Σ −1 .

LO for Grover inversion • LO circuit for 𝑋 4 −1 1 1 1 4 = 1 1 −1 1 1 𝑋 1 1 −1 1 2 1 1 1 −1

Recursive 𝑋 𝑒 circuit 8 given the circuit for 𝑋 4 and 𝑊 • 𝑋 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 .

𝑒 matrix decomposition 𝑋 • Our scheme implies the factorization 2𝑒 = 𝑊 0 𝑒 Φ 𝑊 0 𝑋 0 𝑒 𝑒 𝑒 𝑋 0 𝑊 0 𝑊 0 𝑋 𝑒 𝑒 𝑊 0 𝑒 𝑊 2𝑒 = 𝐼 ⊗ 𝐽 𝑒 0 𝑊 𝑒 2 = 0 1 where 𝑋 2 = 𝐼 . 0 , 𝑊 1

Simulation 8-item Grover search

LO circuit for 𝑄 8 • Prepares an equal superposition state

Simulation results • Fidelity 𝑔 = 〈theo|expt〉 𝑂 = 10 7 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., 𝑉 𝑏 : 𝑡 ↦ 𝑡𝑏 (mod 𝑂) , 0 ≤ 𝑡 < 𝑂 • Given some unitary U what is the minimum number of optical elements needed for its LO circuit?