Eliminating Intermediate Measurements Using Pseudorandom Generators
We show that quantum algorithms of time T and space S ≥ log T with unitary operations and intermediate measurements can be simulated by quantum algorithms of time T ⋅ poly (S) and space {O}(S⋅ log T) with unitary operations and without intermediate measurements. The best results prior to this work required either Ω(T) space (by the deferred measurement principle) or poly(2^S) time [Bill Fefferman and Zachary Remscrim, 2021; Uma Girish et al., 2021]. Our result is thus a time-efficient and space-efficient simulation of algorithms with unitary operations and intermediate measurements by algorithms with unitary operations and without intermediate measurements.
To prove our result, we study pseudorandom generators for quantum space-bounded algorithms. We show that (an instance of) the INW pseudorandom generator for classical space-bounded algorithms [Russell Impagliazzo et al., 1994] also fools quantum space-bounded algorithms. More precisely, we show that for quantum space-bounded algorithms that have access to a read-once tape consisting of random bits, the final state of the algorithm when the random bits are drawn from the uniform distribution is nearly identical to the final state when the random bits are drawn using the INW pseudorandom generator. This result applies to general quantum algorithms which can apply unitary operations, perform intermediate measurements and reset qubits.
quantum algorithms
intermediate measurements
deferred measurement
pseudorandom generator
INW generator
Theory of computation~Pseudorandomness and derandomization
Theory of computation~Quantum complexity theory
76:1-76:18
Regular Paper
Research supported by the Simons Collaboration on Algorithms and Geometry, by a Simons Investigator Award and by the National Science Foundation grants No. CCF-1714779, CCF-2007462.
https://eccc.weizmann.ac.il/report/2021/087/
https://arxiv.org/abs/2106.11877
We thank anonymous reviewers for helpful comments and for pointing out the simulation of intermediate measurements using reset operations.
Uma
Girish
Uma Girish
Princeton University, Princeton, NJ, USA
http://cs.princeton.edu/~ugirish
Ran
Raz
Ran Raz
Princeton University, Princeton, NJ, USA
https://www.wisdom.weizmann.ac.il/~/ranraz/
10.4230/LIPIcs.ITCS.2022.76
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Uma Girish and Ran Raz
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