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Elementary gates for quantum computation
Open accessAbstract
We show that a set of gates that consists of all one-bit quantum gates [U(2)] and the two-bit exclusive-OR gate [that maps Boolean values (x,y) to (x,x\ensuremath{\bigoplus}y)] is universal in the sense that all unitary operations on arbitrarily many bits n [U(${2}^{\mathit{n}}$)] can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical and of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.
Checked 2026-10-06 · 9 references not checked yet
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TL;DR
The paper proves that all one-bit quantum gates together with the two-bit exclusive-or gate are universal for quantum computation, and derives explicit gate constructions and bounds for generalized Toffoli and arbitrary unitary operations. It gives polynomial-size constructions for many important gates, including Θ(n²) basic operations for general ∧_{n−1}(U) gates and Θ(n³4ⁿ) two-bit gates for arbitrary n-bit unitaries.
Key findings
- The set consisting of all one-bit gates U(2) and the two-bit XOR gate is universal: every unitary operation on n bits, U(2ⁿ), can be expressed as a composition of these gates.
- A ∧₁(U) gate can be simulated by at most six basic gates: four one-bit gates and two XOR gates (Corollary 5.3). A ∧₂(U) gate can be simulated by at most sixteen basic gates: eight one-bit gates and eight XOR gates (Corollary 6.2).
- For n ≥ 7, a ∧_{n−2}(σ_x) gate can be simulated by 8(n−5) three-bit Toffoli gates, or by 48n−204 basic operations (Corollary 7.4).
- General ∧_{n−1}(U) gates can be simulated in Θ(n²) basic operations, specifically 48n² + O(n) (Corollary 7.6), with an Ω(n) lower bound for nonscalar ∧_{n−1}(U) (Lemma 7.7). Approximation within ε is possible in Θ(n log(1/ε)) basic operations (Lemma 7.8).
- Special cases are more efficient: for W ∈ SU(2), ∧_{n−2}(W) can be simulated in Θ(n) basic operations (Corollary 7.10), and with one fixed workspace bit, general ∧_{n−2}(U) also takes Θ(n) basic operations (Corollary 7.12).
- Any unitary operator on n bits can be simulated exactly using Θ(n³4ⁿ) two-bit gates and no work bits, via a Reck-type decomposition (Section 8).
Methods
- Theoretical quantum gate-array formalism: defines ∧_m(U) gates, Toffoli gates, and the two-bit XOR gate, then constructs networks from basic operations consisting of one-bit gates ∧₀(U) and the XOR gate ∧₁(σ_x).
- Proofs use matrix factorizations of 2×2 unitaries, SU(2) identities, recursive gate constructions, grey-code sequences, gate merging, phase-congruence arguments, and asymptotic recurrence/lower-bound analysis.
- No physical experiments are reported; the paper mentions numerical study as evidence that some constructions are minimal for small cases (Section 8).
Limitations
- The paper explicitly does not address physical realization difficulties, stating that error correction and quantum coherence are “very serious” problems and referring the reader elsewhere.
- It states that for most examples there is no proof the constructions are the most economical, though numerical study gives compelling evidence of minimality.
- The exact simulation of arbitrary n-bit unitaries uses Θ(n³4ⁿ) two-bit gates, which is exponential; the paper notes this is unlikely to provide a polynomial-time simulation for the special unitaries of most interest in quantum computation, and that a truly efficient design methodology remains to be found.
- The Ω(n) lower bound leaves open the possibility of a linear-size simulation of ∧_{n−1}(U) gates, which the paper says it cannot presently show.
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43 No basis given · 10 Not checked yet · 6 Sources not found · 5 Own result · 4 Own interpretation
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In the paper
“Lemma 5.1: For a unitary 2 × 2 matrix W , a ∧ 1 ( W ) gate can be simulated by a network of the form”
- Also stated in: § 5.1 Simulation of General ∧ 1 ( U ) Gates · p. 10 · plain
“where A , B , and C ∈ SU (2) , if and only if W ∈ SU (2) .”
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- Also stated in: § 5.1 Simulation of General ∧ 1 ( U ) Gates · p. 10 · plain
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“Corollary 5.6: For any unitary 2 × 2 matrix U , a ∧ 1 ( U ) gate can be simulated by at most six basic gates: four 1-bit gates ( ∧ 0 ), and two gates ( ∧ 1 ( V ) ), where V is of the form V = R z ( α ) · R y ( θ ) · R z ( α ) · σ x .”
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“Still, the presence of a workspace plays an important role in the quantum gate constructions - we find that to implement a family of ∧ m gates exactly, the time required for our implementation can be reduced from Θ( m 2 ) to Θ( m ) merely by the introduction of one bit for workspace.”
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“Lemma 6.1: For any unitary 2 × 2 matrix U , a ∧ 2 ( U ) gate can be simulated by a network of the form”
- Also stated in: § 6.1 Simulation of General ∧ 2 ( U ) Gates · p. 14 · plain
“where V is unitary.”
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- Also stated in: § 6.1 Simulation of General ∧ 2 ( U ) Gates · p. 14 · plain
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“Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form”
- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 12 · plain
“where A and B ∈ SU (2) if and only if W is of the form”
- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 12 · plain
“where α and θ are real-valued.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 12 · plain
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In the paper
“Corollary 6.2: For any unitary 2 × 2 matrix U , a ∧ 2 ( U ) gate can be simulated by at most sixteen basic gates: eight 1-bit gates ( ∧ 0 ) and eight XOR gates ( ∧ 1 ( σ x ) ).”
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“We will also exhibit a number of explicit constructions of ∧ m ( U ) using ∧ 1 ( U ), which can all be made polynomial in m .”
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“Lemma 4.3: For any special unitary matrix W ( W ∈ SU (2) ), there exist matrices A , B , and C ∈ SU (2) such that A · B · C = I and A · σ x · B · σ x · C = W .”
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“Corollary 5.3: For any unitary 2 × 2 matrix U , a ∧ 1 ( U ) gate can be simulated by at most six basic gates: four 1-bit gates ( ∧ 0 ), and two XOR gates ( ∧ 1 ( σ x ) ).”
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“Examples of matrices of the form of Lemma 5.5 are the Pauli matrices”
- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 13 · plain
“(as well as σ x itself).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 13 · plain
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“Lemma 5.5: A ∧ 1 ( V ) gate can be simulated by a construction of the form”
- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 13 · plain
“where A and B are unitary if and only if V is of the form”
- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 12 · plain
“where α and θ are real-valued.”
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- Also stated in: § Lemma 5.4: A ∧ 1 ( W ) gate can be simulated by a network of the form · p. 13 · plain
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In the paper
“A particular feature of the ∧ 1 ( σ z ) gates is that they are symmetric with respect to their input bits.”
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“Examples of matrices of the form of Lemma 5.4 are R y ( θ ) itself, as well as R z ( α ) = R z ( α 2 ) · R y (0) · R z ( α 2 ). However, R x ( θ ) is not of this form.”
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“Lemma 5.2: For any δ and S = Ph ( δ ) , a ∧ 1 ( S ) gate can be simulated by a network of the form”
- Also stated in: § 5.1 Simulation of General ∧ 1 ( U ) Gates · p. 11 · plain
“where E is unitary.”
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- Also stated in: § 5.1 Simulation of General ∧ 1 ( U ) Gates · p. 11 · plain
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“For example, the ' x -axis rotation matrix' (to use the language suggested by the mapping between SU(2) and SO(3), the group of rigid-body rotations[43])”
- Also stated in: § 5.2 Special Cases · p. 12 · plain
“is special unitary.”
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- Also stated in: § 5.2 Special Cases · p. 12 · plain
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“Lemma 4.1: Every unitary 2 × 2 matrix can be expressed as”
- Also stated in: § 4 Matrix Properties · p. 8 · plain
“where δ , α , θ , and β are real-valued. Moreover, any special unitary 2 × 2 matrix (i.e., with unity determinant) can be expressed as”
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- Also stated in: § 4 Matrix Properties · p. 8 · plain
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“Note that the 'extra' bit above may be reused in the course of several simulations of ∧ m ( U ) gates.”
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In the paper
“The following is a more efficient simulation of a gate congruent to ∧ 2 ( σ x ) modulo phase shifts:”
- Also stated in: § 6.2 Three-bit gates congruent to ∧ 2 ( U ) · p. 16 · plain
“where A = R y ( π 4 ). In the above, the ' ∼ =' indicates that the networks are not identical, but differ at most in the phases of their amplitudes, which are all ± 1 (the phase of the | 101 〉 state is reversed in this case).”
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- Also stated in: § 6.2 Three-bit gates congruent to ∧ 2 ( U ) · p. 16 · plain
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“When the arbitrary two-bit gate is taken as the basic operation, then as we have seen, 5 operations suffice to produce the Toffoli gate (recall Lemma 6.1), 3 produce the Toffoli gate modulo phases (we permit a merging of the operations in the construction of Sec. 6.2), and 13 can be used to produce the 4-bit Toffoli gate (see Lemma 7.1).”
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“Corollary 7.4: On an n -bit network (where n ≥ 7 ), a ∧ n -2 ( σ x ) gate can be simulated by 8( n -5) ∧ 2 ( σ x ) gates (3-bit Toffoli gates), as well as by 48 n -204 basic operations.”
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In the paper
“Note that in this construction and in the ones following, although many of the bits not involved in the gate are operated upon, the gate operation is performed correctly independent of the initial state of the bits (i.e., they do not have to be 'cleared' to 0 first), and they are reset to their initial values after the operations of the gate (as in the computations which occur in [41] and [40]).”
What the sources say
- Not checked yetcites [41] [41] Michael Ben-Or and Richard Cleve, 'Computing algebraic formulas using a constant number of registers', SIAM J. Comput. 21 , 54 (1992). -
- Not checked yetcites [40] [40] Richard Cleve, 'Reversible Programs and Simple Product Ciphers' in Methodologies for Designing Block Ciphers and Cryptographic Protocols , PhD Thesis, University of Toronto, pp. 2-56 (1989). -
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“Later we will use the fact that because ∧ 2 ( σ x ) is its own inverse, either the simulation of Lemma 6.1 or the time-reversed simulation (in which the order of the gates is reversed, and each unitary operator is replaced by its Hermitian conjugate) may be used.”
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“The formalism we use for quantum computation, which we call a quantum 'gate array' was introduced by Deutsch [24], who showed that a simple generalization of the Toffoli gate (the three-bit gate ∧ 2 ( R x ), in the language introduced later in this paper) suffices as a universal gate for quantum computing.”
What the sources say
- Not matchedcites [24] [24] D. Deutsch, 'Quantum computational networks', Proc. Roy. Soc. Lond. A 425 , 73 (1989). -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“It is possible to perform any computation in a way that is reversible both logically -i.e., the computation is a sequence of bijective transformations-and thermodynamically -the computation could in principle be performed by a physical apparatus dissipating arbitrarily little energy [2].”
What the sources say
- Not checked yetcites [2] [2] R. Landauer, 'Irreversibility and heat generation in the computing process', IBM J. Res. Dev. 5 , 183 (1961); C. H. Bennett, 'Logical reversibility of computation', IBM Journal of Research and Development, 17 , 525 (1973); Yves Lecerf, 'Machines de Turing reversibles. Recursive insolubilite en nεN de l'equation u = θ n ou θ est un 'isomorphism de codes'. Comptes Rendus 257 , 2597-2600 (1963); C. H. Bennett, 'Time/space trade-offs for reversible computation', SIAM J. Comput. 18 , 766 (1989). For a review, see C. H. Bennett and R. Landauer, 'Physical limits of computation', Scientific Americ source
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“Fortunately, Yao has shown [27] that acyclic quantum gate arrays can simulate quantum Turing machines.”
What the sources say
- Not checked yetcites [27] [27] A. C.-C. Yao, 'Quantum Circuit Complexity', Proceedings of the 34th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1993), p. 352. -
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In the paper
“( R x is of special interest because ∧ 2 ( iR x ) is the 'Deutsch gate'[24], which was shown to be universal for quantum logic.)”
What the sources say
- Not matchedcites [24] [24] D. Deutsch, 'Quantum computational networks', Proc. Roy. Soc. Lond. A 425 , 73 (1989). -
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In the paper
“Sleator and Weinfurter[9], who exhibited a scheme for obtaining the Toffoli gate with a sequence of exactly five two-bit gates.”
What the sources say
- Not matchedcites [9] [9] Tycho Sleator and Harald Weinfurter, 'Realizable universal quantum logic gates', preprint (1994). -
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“In fact, it is well known that only a tiny fraction of Boolean functions (those which are linear with respect to modulo 2 arithmetic) can be generated with these gates[39].”
What the sources say
- Not checked yetcites [39] [39] D. Coppersmith and E. Grossman, 'Generators for Certain Alternating Groups with Applications to Cryptography', SIAM J. Applied Math. 29 , No. 4, pp. 624-627 (1975). -
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“Fredkin and Toffoli[3] showed that there exists a 3-bit 'universal gate' for reversible computation, that is, a gate which, when applied in succession to different triplets of bits in a gate array, could be used to simulate any arbitrary reversible computation.”
What the sources say
- Not matchedcites [3] [3] E. Fredkin and T. Toffoli, 'Conservative Logic', Internat. J. Theoret. Phys. 21 , 219 (1982). -
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In the paper
“This unitary evolution on a normalized state vector is known to be the correct physical description of an isolated system evolving in time according to the laws of quantum mechanics[1].”
What the sources say
- Not checked yetcites [1] [1] P. A. M. Dirac, The Principles of Quantum Mechanics , (Oxford, 1958), Chap. 5; for a modern treatment, see A. Peres, Quantum Theory: Concepts and Methods , (Kluwer, 1993), Chap. 8.6. - source
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“In fact for general k, time-efficient design as in Fig. 5a requires k-2 zeroed work qubits within depth 2k-3, whereas space-efficient design as in Fig. 5b uses only one arbitrary qubit within depth 8k-24 (for k ≥ 5) [32].”
From Time–space complexity of quantum search algorithms in symmetric cryptanalysis: applying to AES and SHA-2 (2018) · cites this paper as [32]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Any two-qubit gate can be obtained from this minimal set by using circuit decompositions (Barenco et al. , 1995; Blaauboer and De Visser, 2008; de Guise et al. , 2018; Peterson et al. , 2020; Watts et al. , 2013).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Barenco et al., 1995
In the citing paper
“We should point out here that the results presented here give faster gates than any results for the maximum speed of Toffoli gates in the literature [2,3].”
From Speed limits for quantum gates in multiqubit systems (2012) · cites this paper as [3]
In the citing paper
“In particular, one can compare the factor 1.9 in Table II with the corresponding factor of 2.2 in Ref. [3]: an improvement of about 15%.”
From Speed limits for quantum gates in multiqubit systems (2012) · cites this paper as [3]
In the citing paper
“An optimal-control-theory-related calculation based on certain forms of pulses has found a gate time of about 2.2 times the CNOT gate time for qubits coupled in a triangle geometry [3].”
From Speed limits for quantum gates in multiqubit systems (2012) · cites this paper as [3]
In the citing paper
“Although computational universality might at first seem to be a stringent demand on a physical system, a wide variety of physical systems - ranging from nearest neighbor Ising models 52 to quantum electrodynamics 84 and conformal field theories 86 - are known to be computationally universal 51 -53 , 55 -65 .”
From Ultimate physical limits to computation (2000) · cites this paper as [59]
In the citing paper
“In other words, given an arbitrary quantum gate, there is a circuit built from one-and two-qubit gates that can enact it to arbitrary precision.”
From The prospects of quantum computing in computational molecular biology (2021) · cites this paper as [47]
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Each contradiction between two passages listed here was confirmed by an independent check.
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In the paper
§ 7.1 Linear Simulation of ∧ n -2 ( σ x ) Gates on n -Bit Networks · p. 20
“a ∧ n -2 ( σ x ) gate can be simulated by 8( n -5) ∧ 2 ( σ x ) gates”
§ 7.1 Linear Simulation of ∧ n -2 ( σ x ) Gates on n -Bit Networks · p. 20
“only 4 of the Toffoli gates, the ones which involve the last bit in the diagram above, need to be simulated exactly according to the construction of Corollary 6.2. Thus these 4 gates are simulated by 16 basic operations, while the other 8 n -36 Toffoli gates are simulated in just 6 basic operations.”
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Metadata
- Authors
- Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. A., & Weinfurter, H.
- Affiliations
- AT&T (United States) (US)
- ORCID
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1995-11-01
- Volume
- 52
- Issue
- 5
- Pages
- 3457-3467
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2102008187
- Semantic Scholar
- 59b447f58246fbbdffd5e896f83a3a142eca5cf1
- Cited by
- 4,311
- References
- 52
- Citations by year
- 2026: 201 · 2025: 271 · 2024: 253 · 2023: 271 · 2022: 258 · 2021: 218 · 2020: 200 · 2019: 179 · 2018: 152 · 2017: 125 · 2016: 125 · 2015: 130 · 2014: 119 · 2013: 124 · 2012: 136
- FWCI
- 63.39
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Quantum-Dot Cellular Automata
- Keywords
- quantum gates, XOR gate, gate decomposition, circuit complexity, lower bounds, upper bounds, quantum computing
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