nLab deferred measurement principle

Idea

In quantum physics and specifically in quantum information theory, the principle of deferred measurement is a theorem which says that

• any quantum circuit involving quantum measurement of some of the qbits followed by quantum gates controlled by the respective measurement outcomes

is equivalent (equal as a function from given input to output data types) as

• a quantum circuit in which all quantum measurement happens “at the end”, i.e. where no quantum gates are classically controlled by previous measurement results, but all quantumly controlled by coherent control qbits.

The principle can be useful in practice for optimizing quantum circuits. It also clearly relates to the issue of interpretations of quantum mechanics: Since it is the collapse of the wavefunction upon quantum measurement which makes the interpretation of quantum mechanics subtle, it is interesting to note that this collapse may be (arbitrarily) deferred, in a precise sense.

Formalizations

See Gurevich & Blass 2021 for a general formalization and proof.

As an axiom for syntax of a quantum programming language: Staton (2015), Axiom B (p. 6 of 12).

Alternatively: In terms of the discussion at quantum circuits via dependent linear types, the deferred measurement principle is essentially the Kleisli equivalence for the necessity comonad $\Box_B$ on dependent linear types, like this:

Related concepts

• no-cloning theorem

References

While the principle of deferred measurement is a classical statement in quantum information theory, it was not defined or proven in generality and with precision (hence has remained folklore) until Gurevich & Blass 2021 (see the critical discussion of the literature provided there).

Accounts of the informal statement:

• Michael A. Nielsen, Isaac L. Chuang, §4.4 of Quantum computation and quantum information, Cambridge University Press (2000) [doi:10.1017/CBO9780511976667, pdf, pdf]

As an axiom for quantum programming languages:

• Sam Staton, Axiom B (p. 6) in: Algebraic Effects, Linearity, and Quantum Programming Languages, POPL ‘15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2015) [doi:10.1145/2676726.2676999, pdf]

The above discussion and graphics follows:

• CQTS, Quantum Data Types via Linear HoTT (Nov 2022)

See also:

• Wikipedia, Deferred Measurement Principle

General precise statement and proof:

• Yuri Gurevich, Andreas Blass, Quantum circuits with classical channels and the principle of deferred measurements, Theoretical Computer Science 920 (2022) 21–32 [arXiv:2107.08324, doi:10.1016/j.tcs.2022.02.002]