In quantum information and computation, the Solovay–Kitaev theorem says that if a set of single-qubitquantum gates generates a densesubgroup of SU(2), then that set can be used to approximate any desired quantum gate with a short sequence of gates that can also be found efficiently. This theorem is considered one of the most significant results in the field of quantum computation and was first announced by Robert M. Solovay in 1995 and independently proven by Alexei Kitaev in 1997.[1][2]Michael Nielsen and Christopher M. Dawson have noted its importance in the field.[3]
A consequence of this theorem is that a quantum circuit of constant-qubit gates can be approximated to error (in operator norm) by a quantum circuit of gates from a desired finite universal gate set.[4] By comparison, just knowing that a gate set is universal only implies that constant-qubit gates can be approximated by a finite circuit from the gate set, with no bound on its length. So, the Solovay–Kitaev theorem shows that this approximation can be made surprisingly efficient, thereby justifying that quantum computers need only implement a finite number of gates to gain the full power of quantum computation.
Let be a finite set of elements in SU(2) containing its own inverses (so implies ) and such that the group they generate is dense in SU(2). Consider some . Then there is a constant such that for any , there is a sequence of gates from of length such that . That is, approximates to operator norm error.[3] Furthermore, there is an efficient algorithm to find such a sequence. More generally, the theorem also holds in SU(d) for any fixed d.
This theorem also holds without the assumption that contains its own inverses, although presently with a larger value of that also increases with the dimension .[5]
Quantitative boundsedit
The constant can be made to be for any fixed .[6] However, there exist particular gate sets for which we can take , which makes the length of the gate sequence optimal up to a constant factor.[7]
Proof ideaedit
Every known proof of the fully general Solovay–Kitaev theorem proceeds by recursively constructing a gate sequence giving increasingly good approximations to .[3] Suppose we have an approximation such that . Our goal is to find a sequence of gates approximating to error, for . By concatenating this sequence of gates with , we get a sequence of gates such that .
The main idea in the original argument of Solovay and Kitaev is that commutators of elements close to the identity can be approximated "better-than-expected". Specifically, for satisfying and and approximations satisfying and , then
where the big O notation hides higher-order terms. One can naively bound the above expression to be , but the group commutator structure creates substantial error cancellation.
We can use this observation to approximate as a group commutator . This can be done such that both and are close to the identity (since ). So, if we recursively compute gate sequences approximating and to error, we get a gate sequence approximating to the desired better precision with . We can get a base case approximation with constant with an exhaustive search of bounded-length gate sequences.
^Kitaev, Alexei Yu.; Shen, Alexander; Vyalyi, Mikhail N. (2002). Classical and quantum computation. Providence, Rhode Island: American Mathematical Society. ISBN0-8218-2161-X. OCLC 48965167.
^ abcDawson, Christopher M.; Nielsen, Michael (2006-01-01). "The Solovay-Kitaev algorithm". Quantum Information & Computation. 6: 81–95. arXiv:quant-ph/0505030. doi:10.26421/QIC6.1-6.
^Nielsen, Michael A.; Chuang, Isaac L. (2010). "The Solovay–Kitaev theorem". Quantum Computation and Quantum Information: 10th Anniversary Edition. pp. 617–624. doi:10.1017/cbo9780511976667.019. ISBN9780511976667. Retrieved 2020-05-20.
^Kuperberg, Greg (2023-06-22), "Breaking the cubic barrier in the Solovay-Kitaev algorithm", arXiv:2306.13158 [quant-ph]
^Ross, Neil J.; Selinger, Peter. "Optimal ancilla-free Clifford+T approximation of z-rotations". Quantum Information & Computation. 16 (11–12): 901–953. arXiv:1403.2975. doi:10.26421/QIC16.11-12-1.
April 08, 2024
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In quantum information and computation the Solovay Kitaev theorem says that if a set of single qubit quantum gates generates a dense subgroup of SU 2 then that set can be used to approximate any desired quantum gate with a short sequence of gates that can also be found efficiently This theorem is considered one of the most significant results in the field of quantum computation and was first announced by Robert M Solovay in 1995 and independently proven by Alexei Kitaev in 1997 1 2 Michael Nielsen and Christopher M Dawson have noted its importance in the field 3 A consequence of this theorem is that a quantum circuit of m displaystyle m constant qubit gates can be approximated to e displaystyle varepsilon error in operator norm by a quantum circuit of O mlogc m e displaystyle O m log c m varepsilon gates from a desired finite universal gate set 4 By comparison just knowing that a gate set is universal only implies that constant qubit gates can be approximated by a finite circuit from the gate set with no bound on its length So the Solovay Kitaev theorem shows that this approximation can be made surprisingly efficient thereby justifying that quantum computers need only implement a finite number of gates to gain the full power of quantum computation Contents 1 Statement 2 Quantitative bounds 3 Proof idea 4 ReferencesStatement editLet G displaystyle mathcal G nbsp be a finite set of elements in SU 2 containing its own inverses so g G displaystyle g in mathcal G nbsp implies g 1 G displaystyle g 1 in mathcal G nbsp and such that the group G displaystyle langle mathcal G rangle nbsp they generate is dense in SU 2 Consider some e gt 0 displaystyle varepsilon gt 0 nbsp Then there is a constant c displaystyle c nbsp such that for any U SU 2 displaystyle U in mathrm SU 2 nbsp there is a sequence S displaystyle S nbsp of gates from G displaystyle mathcal G nbsp of length O logc 1 e displaystyle O log c 1 varepsilon nbsp such that S U e displaystyle S U leq varepsilon nbsp That is S displaystyle S nbsp approximates U displaystyle U nbsp to operator norm error 3 Furthermore there is an efficient algorithm to find such a sequence More generally the theorem also holds in SU d for any fixed d This theorem also holds without the assumption that G displaystyle mathcal G nbsp contains its own inverses although presently with a larger value of c displaystyle c nbsp that also increases with the dimension d displaystyle d nbsp 5 Quantitative bounds editThe constant c displaystyle c nbsp can be made to be log 1 5 2 2 d 1 44042 d displaystyle log 1 sqrt 5 2 2 delta 1 44042 ldots delta nbsp for any fixed d gt 0 displaystyle delta gt 0 nbsp 6 However there exist particular gate sets for which we can take c 1 displaystyle c 1 nbsp which makes the length of the gate sequence optimal up to a constant factor 7 Proof idea editEvery known proof of the fully general Solovay Kitaev theorem proceeds by recursively constructing a gate sequence giving increasingly good approximations to U SU 2 displaystyle U in operatorname SU 2 nbsp 3 Suppose we have an approximation Un 1 SU 2 displaystyle U n 1 in operatorname SU 2 nbsp such that U Un 1 en 1 displaystyle U U n 1 leq varepsilon n 1 nbsp Our goal is to find a sequence of gates approximating UUn 1 1 displaystyle UU n 1 1 nbsp to en displaystyle varepsilon n nbsp error for en lt en 1 displaystyle varepsilon n lt varepsilon n 1 nbsp By concatenating this sequence of gates with Un 1 displaystyle U n 1 nbsp we get a sequence of gates Un displaystyle U n nbsp such that U Un en displaystyle U U n leq varepsilon n nbsp The main idea in the original argument of Solovay and Kitaev is that commutators of elements close to the identity can be approximated better than expected Specifically for V W SU 2 displaystyle V W in operatorname SU 2 nbsp satisfying V I d1 displaystyle V I leq delta 1 nbsp and W I d1 displaystyle W I leq delta 1 nbsp and approximations V W SU 2 displaystyle tilde V tilde W in operatorname SU 2 nbsp satisfying V V d2 displaystyle V tilde V leq delta 2 nbsp and W W d2 displaystyle W tilde W leq delta 2 nbsp then VWV 1W 1 V W V 1W 1 O d1d2 displaystyle VWV 1 W 1 tilde V tilde W tilde V 1 tilde W 1 leq O delta 1 delta 2 nbsp where the big O notation hides higher order terms One can naively bound the above expression to be O d2 displaystyle O delta 2 nbsp but the group commutator structure creates substantial error cancellation We can use this observation to approximate UUn 1 1 displaystyle UU n 1 1 nbsp as a group commutator Vn 1Wn 1Vn 1 1Wn 1 1 displaystyle V n 1 W n 1 V n 1 1 W n 1 1 nbsp This can be done such that both Vn 1 displaystyle V n 1 nbsp and Wn 1 displaystyle W n 1 nbsp are close to the identity since UUn 1 1 I en 1 displaystyle UU n 1 1 I leq varepsilon n 1 nbsp So if we recursively compute gate sequences approximating Vn 1 displaystyle V n 1 nbsp and Wn 1 displaystyle W n 1 nbsp to en 1 displaystyle varepsilon n 1 nbsp error we get a gate sequence approximating UUn 1 1 displaystyle UU n 1 1 nbsp to the desired better precision en displaystyle varepsilon n nbsp with en displaystyle varepsilon n nbsp We can get a base case approximation with constant e0 displaystyle varepsilon 0 nbsp with an exhaustive search of bounded length gate sequences References edit Kitaev A Yu 1997 12 31 Quantum computations algorithms and error correction Russian Mathematical Surveys 52 6 1191 1249 Bibcode 1997RuMaS 52 1191K doi 10 1070 rm1997v052n06abeh002155 ISSN 0036 0279 S2CID 250816585 Kitaev Alexei Yu Shen Alexander Vyalyi Mikhail N 2002 Classical and quantum computation Providence Rhode Island American Mathematical Society ISBN 0 8218 2161 X OCLC 48965167 a b c Dawson Christopher M Nielsen Michael 2006 01 01 The Solovay Kitaev algorithm Quantum Information amp Computation 6 81 95 arXiv quant ph 0505030 doi 10 26421 QIC6 1 6 Nielsen Michael A Chuang Isaac L 2010 The Solovay Kitaev theorem Quantum Computation and Quantum Information 10th Anniversary Edition pp 617 624 doi 10 1017 cbo9780511976667 019 ISBN 9780511976667 Retrieved 2020 05 20 Bouland Adam Giurgica Tiron Tudor 2021 12 03 Efficient Universal Quantum Compilation An Inverse free Solovay Kitaev Algorithm arXiv 2112 02040 Kuperberg Greg 2023 06 22 Breaking the cubic barrier in the Solovay Kitaev algorithm arXiv 2306 13158 quant ph Ross Neil J Selinger Peter Optimal ancilla free Clifford T approximation of z rotations Quantum Information amp Computation 16 11 12 901 953 arXiv 1403 2975 doi 10 26421 QIC16 11 12 1 Retrieved from https en wikipedia org w index php title Solovay Kitaev theorem amp oldid 1196178613, wikipedia, wiki, book, books, library,