Quantum state engineering using one-dimensional discrete-time quantum walks Luca Innocenti, Helena Majury, Mauro Paternostro and Alessandro Ferraro Centre for Theoretical Atomic, Molecular, and Optical Physics, chool of Mathematics and Physics, Queen s University Belfast, Queen s University Belfast Taira Giordani, Nicolò pagnolo and Fabio ciarrino Dipartimento di Fisica, apienza Università di Roma www.quantumlab.it QTML 2017, Verona 7/11/2017
Framework Quantum state engineering Quantum computation and quantum simulation Quantum walks Jacob Biamonte et al., Quantum Machine Learning, Nature 549, (2017)
Framework Quantum state engineering Quantum computation and quantum simulation Quantum walks Quantum Machine learning Quantum information enhanced by quantum machine learning Jacob Biamonte et al., Quantum Machine Learning, Nature 549, (2017)
Outlook Quantum walks Random walks vs Quantum walks Discrete-time quantum walk on a line Quantum state engineering protocol Reachability conditions and reachable states ingle-photon states and linear optics Linear optics platforms Angular momentum of light Experimental proposal
Quantum Walks Random walks Quantum walks
1-D discrete-time quantum walks Quantum system described by: Walker s position i H P = n n Z Walker s coin s H C =, ψ = i s, u i,s i s
1-D discrete-time quantum walks Quantum walk evolution Coin operator C (k) any unitary transformation in coin space Controlled-shift operator increases or decreases walker s position according to the coin state + ۦ i ۦ = ( i i ) ۦ i ۦ 1 + i + W = C (n) C n 1 C (1)
Quantum state engineering ψ (0) ൿ C (1) C (n 1) C (n) ψ (n) ൿ Coin measurement φ H p Target state ψ (n) ൿ = C (n) ψ (n 1) ൿ = i s, u (n 1) i,s ( i C n s ) The idea is to use 1D discrete quantum walks to produce states in high-dimensional Hilbert space What are the quantum state that can be the result of a quantum walk s routine? L. Innocenti et al., Quantum state engineering using one-dimensional discrete-time quantum walks, arxiv:1710.10518.
1-step reachability conditions ψ (0) ൿ C (1) C (n 1) C (n) ψ (n) ൿ Coin measurement φ H p Target state ψ (0) (0) (0) ൿ = 1 u + u ψ (0) L. Innocenti et al., Quantum state engineering using one-dimensional discrete-time quantum walks, arxiv:1710.10518.
1-step reachability conditions ψ (0) ൿ C (1) C (n 1) C (n) ψ (n) ൿ Coin measurement φ H p Target state (0) ψ (0) ൿ = 1 u 1, (0) + u 1, (1) u 1, (1) u 2, ψ (1) ൿ = 1 u (0) 1, C (1) + u (0) 1, C (1) = (1) (1) = u 1, 1,, +u 2, 2,, ψ (0) (1) (1) u 1, = u2, = 0 L. Innocenti et al., Quantum state engineering using one-dimensional discrete-time quantum walks, arxiv:1710.10518.
2-step reachability conditions ψ (0) ൿ C (1) C (n 1) C (n) ψ (n) ൿ Coin measurement φ H p Target state v 1 (2) v 2 (2) (1) ψ (1) ൿ = u 1, (1) 1,, +u 2, 2, (1) u 1, (1) u 2, v i (n) = (n) u i, (n) u i+1, u (1) 1, = u (1) 3, = 0 ψ (0) v 1 (2) v 2 (2) = 0 L. Innocenti et al., Quantum state engineering using one-dimensional discrete-time quantum walks, arxiv:1710.10518.
General reachability conditions ψ (0) ൿ C (1) C (n 1) C (n) ψ (n) ൿ Coin measurement φ H p Target state u 1, = u n+1, = 0 s v i v n s+i = 0 s = 1, n 1 i=1 The output states of n-step 1D quantum walk satisfy a set of constrains, without any hypotheses over coin operators L. Innocenti et al., Quantum state engineering using one-dimensional discrete-time quantum walks, arxiv:1710.10518.
Reachable states: results ψ (0) ൿ C (1) C (n 1) C (n) ψ (n) ൿ Coin measurement φ H p Target state The Hilbert space of reachable states has dimension 2n Each output state of a discrete time quantum walk has to satisfy a set of constraints Conversely, for each state that verifies these constraints there exists a collection of coin operators that can produce it. φ C (k) L. Innocenti et al., Quantum state engineering using one-dimensional discrete-time quantum walks, arxiv:1710.10518.
Linear optics platforms Integrated linear interferometers Fiber-loop scheme A. Crespi et al., Nature Photonics 7, 322 (2013). Angular momentum of light A. chreiber et al., cience 336, 55 (2012). Bulk interferometric scheme F. Cardano et al., cience Advances 1, e1500087 (2015). M. A. Broome et al., Phys. Rev. Lett. 104, 153602 (2010).
Platforms in lab Integrated linear interferometers A. Crespi et al., Nature Photonics 7, 322 (2013). Angular momentum of light F. Cardano et al., cience Advances 1, e1500087 (2015).
Angular Momentum of Light Light has spin angular momentum (AM) in the polarization degree of freedom R, L Light can carries also orbital angular momentum (OAM) that is encoded in the spatial profile of the wavefront Laguerre-Gauss polinomials
OAM manipulation: Q-plates α r, φ = φq + α 0 Q δ α 0 Q δ α 0 δ L, m = cos 2 δ R, m = cos 2 L, m i sin δ 2 ei2α 0 R, m + 2q R, m i sin δ 2 e i2α 0 L, m 2q V. D Ambrosio et al., Arbitrary, direct and deterministic manipulation of vector beams via electrically-tuned q-plates, cientific Reports 5 (2015)
Experimental proposal Coin operator implemented with sets of motorized waveplates hift operator performed by q-plates Qp R, m L, m 2q, Qp L, m R, m + 2q, Waveplates and polarizing beam splitter to analyze coin state patial light modulators to prepare and measure the walker s position Automatized software controls the various optical elements of the scheme
Experiment on going Coin preparation Q-plates with q=1/2 Type II collinear PDC with PPKTP crystal patial light modulator
Conclusions Novel approach to quantum walk in quantum state engineering framework v 1 (2) v 2 (2) Experimental implementation that exploits orbital angular momentum of light Platform feasible for other applications in quantum state tomography, quantum simulation and quantum machine learning (1) u 1, ψ (0) (1) u 2, L. Innocenti et al., Quantum state engineering using one-dimensional discrete-time quantum walks, arxiv:1710.10518.