Recursive linear optical netwroks for realizing quantum ...tgnm/notes/recQAlg.pdf · APS March Meeting 2016 | 14-18 March 2016. Motivation •Many practical quantum technolgies have

Recursive linear opticalnetworks for realizingquantum algorithms

Gelo Noel Tabia

APS March Meeting 2016 | 14-18 March 2016

Motivation

• Many practical quantum technolgieshave been achieved with linear optics(LO).

• Progress in LO quantum computationwith photonic integrated circuits (PIC)

• Goal: recipes for translating quantumalgorithms into practical LO schemes

Linear optics (LO)

• Photons manipulated by a network of phase shifters and beam splitters

𝑃𝜃 =1 00 𝑒𝑖𝜃

𝜃

𝐵𝜖 =𝜖 1 − 𝜖

1 − 𝜖 − 𝜖

𝜖

Photonic integrated circuit

• Waveguide-based linear optics

Path-encoded qudits

Unitary gates

• Reck, et al. (1994)

• Any unitary 𝑈 ∈ 𝑆𝑈 𝑑 can be realizedby a LO network on 𝑑 modes using𝑑2 − 1 elements

Main results

• Recursive LO networks for quantumFourier transform (QFT) and Groverinversion

• Circuit for 𝑈2𝑑 built using a pair of circuits for 𝑈𝑑

• Unitary matrix decomposition into2 × 2 -block-diagonal matrices

[arXiv:1509.04246]

Quantum Fourier transform

• Fourier transform on quantum states

𝐹4 =1

4

1 11 𝑖

1 1−1 −𝑖

1 −11 −𝑖

1 −1−1 𝑖

Recursive QFT circuit

• e.g. QFT circuit 𝐹8 given circuit for 𝐹4

Γ: 1,2,3,4,5,6,7,8 ↦ (1,3,5,7,2,4,6,8)

Γ

Fourier matrix factorization

• First discovered by Gauss, this is thebasis for fast Fourier transform(Cooley-Tukey algorithm):

𝐹2𝑑 =1

2

𝐼 𝐷𝐷 𝐼

𝐹𝑑 00 𝐹𝑑

Γ

𝐷 = diag(1, 𝜔,… ,𝜔𝑑−1)

𝜔 = 𝑒2𝜋𝑖/𝑑

Grover’s algorithm

• We construct a recursive LO circuit forGrover inversion 𝑊𝑑

𝐺 = 𝑊𝑈𝑓𝑈𝑓: 𝑥 ↦ −1 𝑓(𝑥) 𝑥

Recursive 𝑉𝑑 circuit

• Constructing 𝑉8 from 𝑉4

Recursive 𝑊𝑑 circuit

• Grover inversion 𝑊4

𝑊4 =1

2

−1 11 −1

1 11 1

1 11 1

−1 11 −1

Recursive 𝑊𝑑 circuit

• 𝑊8 given the circuit for 𝑊4 and 𝑉4

Φ

Φ: 1,2,3,4,5,6,7,8 ↦ (5,2,3,4,1,6,7,8)

𝑊𝑑 matrix decomposition

• Formally this corresponds to

𝑊2𝑑 =𝑉𝑑 00 𝑉𝑑

Φ𝑉𝑑 00 𝑉𝑑

𝑊𝑑 00 𝑊𝑑

𝑉2𝑑 = 𝐻⊗ 𝐼𝑑𝑉𝑑 00 𝑉𝑑

𝑉2 = 𝐻 = 1

2

1 11 −1

𝑊2 =0 11 0

Simulation results

• Haar-uniform input states for QFT

𝑁 = 107

𝜇 = 0.861𝜎 = 0.056

Error model• BS: 𝜖 ~ 𝑁(0.5, 0.04 )• PS: 𝛼 ~ 𝑁+(0.05, 0.025)

𝑓 = 𝜙 𝜓 2

Simulation results

• 8-item Grover search

𝑁 = 107

𝜇 = 0.762𝜎 = 0.099

𝑓 = 𝜙 𝜓 2

Error model• BS: 𝜖 ~ 𝑁(0.5, 0.04 )• PS: 𝛼 ~ 𝑁+(0.05, 0.025)

Conclusion

• Recursive formula for LO circuits of QFT and Grover inversion

• Size complexity: 𝑑 log 𝑑 ; 𝑑2 𝑑2

• Depth complexity: 5𝑑 2𝑑

• Applications: tool for boson sampling(QFT), Grover-like algorithms (GI)

Acknowledgement

• Quantum cryptography group in Tartu

DominiqueUnruh

o Post-quantum security of encryption schemeso Verification of quantum cryptographic proofso Quantum collision finding problemo Quantum proofs of knowledge

EhsanEhbrami

MayureshAnand

Tore Vincent Carstens