Quantum signal processing Aram Harrow UW Computer Science & Engineering

probabilistic bits description: evolution: q 1-q 1-r r stochastic matrix measurement: 0 1 with probability p 0 with probability p 1

quantum bits (qubits) description: evolution: u 00 u 01 u 11 u10 unitary matrix measurement: 0 1 with probability |a 0 | 2 with probability |a 1 | 2

interference

signal processing? 1. Can quantum devices provide hardware or software improvements to signal processing? 2. Can quantum-inspired math help inform signal processing on existing devices?

hardware improvements Processing single photons/electrons/phonons is naturally quantum. Entanglement-assisted metrology often offers square-root advantages, although not always in a way that is robust to noise. Less obvious: longer-baseline telescopes using quantum repeaters. [Gottesman et al., arXiv: ]

software improvements Grover’s algorithm: Search N possibilities in time O(N 1/2 ). Shor’s algorithm: Factor a log(N)-digit number in time poly(log(N)). Based on the quantum Fourier transform. If for x=0,…,N-1, then a quantum computer can efficiently sample from, where Superpositions of {0,…,N-1} require log(N) qubits.

Large linear systems Input: Assume A is s-sparse and has condition number κ. Output: x such that Ax=b Classically: Iterative methods output x in time O(κ N s log(1/ε)). A quantum computer: Can produce a state with amplitudes proportional to x in time O(κ log(N) s 4 / ε). [H-Hassidim-Lloyd, Phys. Rev. Lett ‘09] [Ambainis, arXiv: ]

Challenges Knowing what to speed up Scope of quantum speedups is unknown Exponential speedups require problems with small input and output descriptions. Linearity and symmetry may play a role.

quantum-inspired math? Eldar & Oppenheim use formalism of quantum measurement to devise new signal-processing techniques. Tensor optimization problem: Given an n×n×n array A ijk, maximize |∑ ijk A ijk x i y j z k | over unit vectors x,y,z.

more reading General quantum information background: M.A. Nielsen and I.L. Chuang. “Quantum Computation and Quantum Information.” CUP J. Preskill. Signal processing using quantum formalism: Y.C. Eldar and A.V. Oppenheim, “Quantum Signal Processing,” Signal Processing Mag., vol. 19, pp , Nov My work: Linear systems: arxiv.org/ Tensor optimization: arxiv.org/