Deferred measurement principle

Two equivalent quantum logic circuits. One where measurement happens first, and one where an operation conditioned on the to-be-measured qubit happens first.

Measurement is performed early and the resulting classical bits are sent. The classical bits control if the 1-qubit X and Z gates are executed, allowing teleportation. ^[1]

By moving the measurement to the end, the 2-qubit controlled-X and -Z gates need to be applied, which requires both qubits to be near (i.e. at a distance where 2-qubit quantum effects can be controlled), and thus limits the distance of the teleportation. While logically equivalent, deferring the measurement have physical implications.

Example: Two variants of the teleportation circuit. The 2-qubit states ${\displaystyle |\Phi ^{+}\rangle }$ and ${\displaystyle |\beta _{00}\rangle }$ refer to the same Bell state.

The deferred measurement principle is a result in quantum computing which states that delaying measurements until the end of a quantum computation doesn't affect the probability distribution of outcomes. ^{[2] [3]}

A consequence of the deferred measurement principle is that measuring commutes with conditioning. The choice of whether to measure a qubit before, after, or during an operation conditioned on that qubit will have no observable effect on a circuit's final expected results.

Thanks to the deferred measurement principle, measurements in a quantum circuit can often be shifted around so they happen at better times. For example, measuring qubits as early as possible can reduce the maximum number of simultaneously stored qubits; potentially enabling an algorithm to be run on a smaller quantum computer or to be simulated more efficiently. Alternatively, deferring all measurements until the end of circuits allows them to be analyzed using only pure states.

In error corrected code blocks, the instruction set is limited to some discrete non-universal set. ^[4] This error corrected instruction set almost always include the Pauli gates (because the pauli group describes the observable effects of single-qubit effects of decoherence). One way to extend the error corrected instruction set is to exploit the principle of deferred measurement, to convert quantum controlled Pauli gates that are typically not in the error corrected instruction set, into classically controlled Pauli gates, that are. Since the receiving end of the teleportation circuit is just Pauli gates, it can be used to inject gates into the corrected code block ^[5] and in some cases, depending on the error code used, allows to extend the available instruction set to a universal set.

See also

• Holevo's theorem
• One-way quantum computer
• Quantum gate teleportation

References

1. Nielsen, Michael A.; Chuang, Isaac (2010). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. pp. 26–28. ISBN   978-1-10700-217-3. OCLC   43641333.
2. Michael A. Nielsen; Isaac L. Chuang (9 December 2010). "4.4 Measurement". Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press. p. 186. ISBN   978-1-139-49548-6.
3. Odel A. Cross (5 November 2012). "5.2.2 Deferred Measurement". Topics in Quantum Computing. O. A. Cross. p. 348. ISBN   978-1-4800-2749-7.
4. Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN   978-1-84628-887-6.
5. Gottesman, Daniel; Chuang, Isaac L. (1999). "Quantum Teleportation is a Universal Computational Primitive". Nature . 402 (6760): 390–393. arXiv: quant-ph/9908010 . Bibcode:1999Natur.402..390G. doi:10.1038/46503. S2CID   4411647.