quantum signal processing aram harrow uw computer science & engineering [email protected]
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probabilistic bits
description:
evolution:
0
1
0
1
q
1-q
1-rr stochastic matrix
measurement:0
1
with probability p0
with probability p1
quantum bits (qubits)
description:
evolution:
0
1
0
1
u00
u01
u11
u10 unitary matrix
measurement:0
1
with probability |a0|2
with probability |a1|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:1107.2939]
software improvementsGrover’s algorithm: Search N possibilities in
time O(N1/2).
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 aquantum computer can efficiently sample from , where
Superpositions of {0,…,N-1} require log(N) qubits.
Large linear systemsInput:
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) s4 / ε).[H-Hassidim-Lloyd, Phys. Rev. Lett ‘09][Ambainis, arXiv:1010.4458]
ChallengesKnowing 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 Aijk, maximize |∑ijk Aijk xi yj zk| 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 2000.
J. Preskill. www.theory.caltech.edu/~preskill/ph229/
Signal processing using quantum formalism:
Y.C. Eldar and A.V. Oppenheim, “Quantum Signal Processing,” Signal Processing Mag., vol. 19, pp. 12-32, Nov 2002.
My work:
Linear systems: arxiv.org/0811.3171
Tensor optimization: arxiv.org/1001.0017