Circuit implementing the swap test between two states and
The swap test is a procedure in quantum computation that is used to check how much two quantum states differ, appearing first in the work of Barenco et al.[1]
and later rediscovered by Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf.[2] It appears commonly in quantum machine learning, and is a circuit used for proofs-of-concept in implementations of quantum computers.[3][4]
Formally, the swap test takes two input states and and outputs a Bernoulli random variable that is 1 with probability (where the expressions here use bra–ket notation). This allows one to, for example, estimate the squared inner product between the two states, , to additive error by taking the average over runs of the swap test.[5] This requires copies of the input states. The squared inner product roughly measures "overlap" between the two states, and can be used in linear-algebraic applications, including clustering quantum states.[6]
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Adriano Barenco, André Berthiaume, David Deutsch, Artur Ekert, Richard Jozsa, Chiara Macchiavello (1997). "Stabilization of Quantum Computations by Symmetrization". SIAM Journal on Computing. 26 (5): 1541–1557. arXiv:quant-ph/9604028. doi:10.1137/S0097539796302452.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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Harry Buhrman, Richard Cleve, John Watrous, Ronald de Wolf (2001). "Quantum Fingerprinting". Physical Review Letters. 87 (16): 167902. arXiv:quant-ph/0102001. Bibcode:2001PhRvL..87p7902B. doi:10.1103/PhysRevLett.87.167902. PMID 11690244. S2CID 1096490.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^Kang Min-Sung, Heo Jino, Choi Seong-Gon, Moon Sung, Han Sang-Wook (2019). "Implementation of SWAP test for two unknown states in photons via cross-Kerr nonlinearities under decoherence effect". Scientific Reports. 9 (1): 6167. Bibcode:2019NatSR...9.6167K. doi:10.1038/s41598-019-42662-4. PMC 6468003. PMID 30992536.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^de Wolf, Ronald (2021-01-20). "Quantum Computing: Lecture Notes". pp. 117–119, 122. arXiv:1907.09415 [quant-ph].
^Wiebe, Nathan; Kapoor, Anish; Svore, Krysta M. (1 March 2015). "Quantum Algorithms for Nearest-Neighbor Methods for Supervised and Unsupervised Learning". Quantum Information and Computation. 15 (3–4). Rinton Press, Incorporated: 316–356. arXiv:1401.2142. doi:10.26421/QIC15.3-4-7. S2CID 37339559.
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