Eliminating Intermediate Measurements in Space-Bounded Quantum Computation

Abstract

A foundational result in the theory of quantum computation known as the "principle of safe storage" shows that it is always possible to take a quantum circuit and produce an equivalent circuit that makes all measurements at the end of the computation. While this procedure is time efficient, meaning that it does not introduce a large overhead in the number of gates, it uses extra ancillary qubits and so is not generally space efficient. It is quite natural to ask whether it is possible to defer measurements to the end of a quantum computation without increasing the number of ancillary qubits.We give an affirmative answer to this question by exhibiting a procedure to eliminate all intermediate measurements that is simultaneously space-efficient and time-efficient. A key component of our approach, which may be of independent interest, involves showing that the wellconditioned versions of many standard linear-algebraic problems may be solved by a quantum computer in less space than seems possible by a classical computer.

Quantum Logarithmic Space and Post-selection

“…If a language is recognized by a polynomial-time logarithmic-space PostPTM M with one-sided bounded-error, then it is also recognized by a polynomial-time logarithmic-space NTM M ′ where M ′ is modified from M such that if M enters the non-postselelecting state, then M ′ enters the rejecting state. By using the same argument, we can also obtain the following result on quantum class PostEQL (note that the first equality comes from NQL = coC = L [16,7]).…”

“…• When A > 1 2 , the probability of observing |− is always greater than |+ in each execution and at least once it is 7 3 times more. Thus, if x ∈ L, the probability of observing all |− 's is at least 7 3 times more than the probability of observing all |+ 's after all executions.…”

“…• When A > 1 2 , the probability of observing |− is always greater than |+ in each execution and at least once it is 7 3 times more. Thus, if x ∈ L, the probability of observing all |− 's is at least 7 3 times more than the probability of observing all |+ 's after all executions. Therefore, after normalizing the final accepting and rejecting postselecting probabilities, it follows that L is recognized by a polynomial-time logarithmic-space postselecting QTM with error bound 3 10 .…”

“…The following relations on logarithmic space quantum and classical complexity classes are already known [9,16,17,7]:…”

“…While Ta-Shma's quantum algorithm used intermediate measurements, a version of this quantum algorithm without measurement was later constructed by Fefferman and Lin [6] (see also [5] for a related result on space-efficient error reduction for unitary quantum computation). Very recent works [7,8] have further showed that many other problems from linear algebra involving wellconditioned matrices can be solved as well in logarithmic space by quantum algorithms, and additionally showed that intermediate measurements can be removed from any space-bounded quantum computation.…”