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Cited by 1,069Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyQuantum Mechanics and Applications

Two-bit gates are universal for quantum computation

Open access
DiVincenzo, David P.·1995·Physical Review A
DOI 10.1103/physreva.51.1015
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Abstract

A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of three-bit gates, by analogy to the universality of the Toffoli three-bit gate of classical reversible computing. Two-bit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins. A ``gearbox quantum computer'' proposed here, based on the principles of atomic-force microscopy, would permit the operation of such two-bit gates in a physical system with very long phase-breaking (i.e., quantum-phase-coherence) times. Simpler versions of the gearbox computer could be used to do experiments on Einstein-Podolsky-Rosen states and related entangled quantum states.

AI snapshot

Written from the paper’s full text.

TL;DR

DiVincenzo proves that two-bit quantum gates are universal for quantum computation, improving on the previous three-bit universality result. The proof uses Lie-group commutator algebra, and the paper also proposes a “gearbox quantum computer” as a possible physical setting for two-bit gates.

Key findings

  • Two-bit gates suffice to generate arbitrary unitary transformations; the paper gives an explicit realization of Deutsch’s three-bit gate \(U_\lambda\) from one- and two-bit gates, with \(U_\lambda\) obtained by four applications of \(S_Q\) and \(\lambda = -2\pi\alpha\).
  • The Lie-algebra proof shows that successive commutation of Hermitian generators of two-bit gates fills out the entire 64-dimensional Lie algebra; four generators \(H_{66}\), \(H_{77}\), \(H^r_{67}\), and \(H^i_{67}\) suffice to produce \(U(8)\).
  • The proposed gearbox uses meshed gears with 16 and 15 teeth, relatively prime so each pair of spins can be brought into contact; nuclear spins may have long dephasing times, estimated at \(10^{10}\) seconds (3 years) under ideal assumptions, compared with an electron-spin upper bound of \(10^{-3}\) seconds and \(\sim 10^{-13}\) seconds for an electron in a gold wire below 1 K.
  • Explicit two-bit sequences can be lengthy: the analog of Fig. 2 for Eqs. (3.12c) and (3.12d) would contain 21 gates, so the construction is not a practical implementation of quantum logic.

Methods

  • Theoretical proof using the commutator algebra of unitary Lie groups; the paper constructs Deutsch’s three-bit gate from one- and two-bit gates via \(8 \times 8\) matrix multiplication and Lie-algebra generators.
  • No experimental data or sample are used; the work is mathematical and includes a conceptual physical proposal for a gearbox quantum computer.
  • Main analysis uses infinitesimal generators, commutation relations, and explicit gate sequences; the error from concatenating small rotations is stated to be of order \(1/\sqrt{N}\).

Limitations

  • The paper states that the proof only provides small rotation angles and lengthy gate sequences, gives no practical design principles for quantum logic, and describes the gearbox as “more thought-provoking than real,” not an immediate experimental proposal.
  • The paper states that no detailed protocol is worked out for inducing arbitrary two-spin unitary operations, and that no current magnetometry is sensitive enough to sense a single proton spin; atomic-scale vibrations and misalignments may be deleterious.
  • Evident from the text: there is no experimental realization or empirical validation, and the long nuclear-spin dephasing estimate is ideal/assumed and “will certainly be much smaller in reality.”

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physreva.51.1015

Cites

23 of 23 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 21 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § B. Proof that Q can be realized by two-bit gates · p. 12

    The equation of Fig. 2 is true to first order in the small parameter delta.

    “to first order in the small parameter δ .”
    Quote in the paperSubstantial claim
  • Findingnone given
    § B. Proof that Q can be realized by two-bit gates · p. 11

    U_lambda is obtained by four applications of S_Q in Eq. (3.1), with the identification lambda = -2 pi alpha.

    “U λ is obtained by four applications of S Q in Eq. (3.1), with the identification λ = -2 πα .”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § C. Completion of the proof: generating the entire Lie algebra · p. 14

    Eqs. (3.12) complete the proof that all necessary three-bit operations can be executed using two-bit gates.

    “Eqs. (3.12) complete the proof that all necessary three-bit operations can be executed using two-bit gates; the explicit sequence of gates can be read off the equations.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Abstract · p. 1

    Quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit, i.e., to generate any arbitrary unitary transformation.

    “A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit.”
    • Also stated in: § I. INTRODUCTION · p. 3 · boosted
      “Here, using the techniques of Lie group theory, I prove the desired new result, that two-bit gates suffice to generate any arbitrary quantum network, i.e., any arbitrary unitary transformation.”
    • Also stated in: § IV. CONCLUSIONS · p. 15 · plain
      “The present work shows that all quantum logic can in principle be designed with two-bit gates; however, it does not offer any practical design principles for quantum logic, and this remains an important open issue for the future.”
    Quote in the paperSubstantial claim
  • Findingown data
    § I. INTRODUCTION · p. 3

    The proof provides an explicit realization of three-bit operations in terms of sequences of two-bit gates.

    “The proof provides an explicit realization of three-bit operations in terms of sequences of two-bit gates, although it remains to be seen whether this may form the basis of a practical, efficient method of designing quantum circuits.”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 21
  • May combine two claims. 11
  • The quote only partly states this claim. 1

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given
    § C. Completion of the proof: generating the entire Lie algebra · p. 14

    Eqs. (3.12) only provide a way of getting unitary operations with small rotation angles.

    “For one thing, Eqs. (3.12) only provide a way of getting unitary operations with small rotation angles.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § B. Proof that Q can be realized by two-bit gates · p. 12

    To obtain U_lambda for any lambda to a desired degree of accuracy, it is only necessary to concatenate a set of small rotations, and the error made by using the set of two-bit operations in Eq. (3.7) is of order 1/sqrt(N).

    “To obtain U λ for any λ to a desired degree of accuracy, it is only necessary to concatenate a set of small rotations, by writing U λ = ( U λ/N ) N ; the error made by using the set of two-bit operations in Eq.(3.7) can be shown to be of order 1 / √ N .”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § C. Completion of the proof: generating the entire Lie algebra · p. 13

    Successive commutation of the Hermitian generators of the set of two-bit gates fills out the entire 64-dimensional Lie algebra spanned by Eqs. (3.9).

    “With this machinery, the question of whether two-bit gates suffice to produce all possible three-bit unitary operations boils down to the question of whether the successive commutation of the Hermitian generators of our set of two-bit gates fills out the entire 64-dimensional Lie algebra spanned by Eqs. (3.9).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning
    § C. Completion of the proof: generating the entire Lie algebra · p. 14

    Eqs. (3.12c) and (3.12d) specify a rather lengthy sequence of two-bit gates, for which the analog of Fig. 2 would contain a sequence of 21 gates.

    “Secondly, these equations specify a rather lengthy sequence of two-bit gates, especially Eqs. (3.12c) and (3.12d), for which the analog of Fig. 2 would contain a sequence of 21 gates!”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § IV. CONCLUSIONS · p. 15

    There is evidence that quantum computers cannot solve the NP class of problems any more rapidly than classical computers, although the random oracle method of proof is not definitive.

    “For example, there is evidence that quantum computers cannot solve the NP class of problems [27] any more rapidly than classical computers, although the method of proof used, employing a so called 'random oracle', is known not to be definitive.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [27] [27] C. H. Bennett, E. Bernstein, G. Brassard, and U. V. Vazirani, 'Strengths and weaknesses of quantum computing', Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, 1994 (to be published). -

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § B. Gearbox quantum computer · p. 7

    Quantum computations under consideration require that the spins be in an initial simple state, e.g., all up, but an assemblage of nuclear spins at any reasonable temperature will typically have a random state drawn from a Boltzmann distribution.

    “Quantum computations under consideration now [5,4] require that the spins be an initial simple state, e.g., all up. But because of the low energy scales involved, an assemblage of nuclear spins at any reasonable temperature will typically have a random state which is drawn from a Boltzmann distribution.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [5] [5] D. R. Simon, 'On the power of quantum computation', Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, 1994 (to be published). -

      Not checked in the free run.

    • cites [4] [4] P. W. Shor, 'Algorithms for quantum computation: discrete log and factoring', Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, 1994 (to be published). -

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § B. Gearbox quantum computer · p. 7

    The output device must sense the state of a single spin and make that information available to the rest of the world, and this can be done only at the end of the quantum computation because it involves strong interaction that destroys unitary evolution.

    “This device must sense the state of a single spin and make that information available to the rest of the world. This operation can be done only at the end of the quantum computation, since it involves strong interaction of the quantum computer with other degrees of freedom, destroying the unitary evolution.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § B. Gearbox quantum computer · p. 8

    In the polymer computer, not just two-bit but also three-bit local operations can be executed, but this is not a crucial advantage.

    “In the polymer computer, not just two-bit but also three-bit local operations can be executed (but this is not a crucial advantagesee the next section).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [15] [15] S. Lloyd, 'A potentially realizable quantum computer', Science, 261 , 1569 (1993); 'Envisioning a quantum supercomputer', Science, 263 , 695 (1994). - source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation
    § B. Gearbox quantum computer · p. 8

    The polymer computer has the disadvantage that the interaction Hamiltonian between the spins can never be turned off, which makes control of the phase of the quantum state particularly ungainly.

    “The polymer computer has the disadvantage that the interaction Hamiltonian between the spins can never be turned off. This does not necessarily lead to insuperable problems, but it makes the control of the phase of the quantum state particularly ungainly.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [15] [15] S. Lloyd, 'A potentially realizable quantum computer', Science, 261 , 1569 (1993); 'Envisioning a quantum supercomputer', Science, 263 , 695 (1994). - source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationreasoning
    § Abstract · p. 1

    A gearbox quantum computer based on atomic force microscopy would permit operation of two-bit gates in a physical system with very long phase breaking, i.e., quantum phase coherence, times.

    “A 'gearbox quantum computer' proposed here, based on the principles of atomic force microscopy, would permit the operation of such two-bit gates in a physical system with very long phase breaking (i.e., quantum phase coherence) times.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: Very long dephasing times for spins can be achieved in the gearbox by making the gear elements magnetically and electronically quiet.
    • Rests on an assumption: Atomic spatial resolution and atomic contact between spins can be achieved in the gearbox.

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

    Interpretationnone given
    § B. Gearbox quantum computer · p. 8

    In the gearbox computer, atomic-scale vibrations and misalignments might be very difficult to control and quite deleterious to operation.

    “In the gearbox computer, atomic-scale vibrations and misalignments might be very difficult to control and quite deleterious to the operation.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § IV. CONCLUSIONS · p. 14

    Quantum computers suffer from instabilities in their time evolution inherent to any Hamiltonian system, and they cannot be error-corrected in any traditional sense because error correction is intrinsically dissipative.

    “As Landauer has discussed [26], quantum computers suffer from instabilities in their time evolution which are inherent to any Hamiltonian system; in addition quantum computers cannot be errorcorrected in any traditional sense, since error correction is intrinsically dissipative.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [26] [26] R. Landauer, 'Is quantum mechanics useful?', Proc. Roy. Soc. Lond., (to be published). -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § B. Gearbox quantum computer · p. 8

    In the polymer computer, unitary operations can go on in parallel, although the ability to address specific pairs of spins is lost.

    “Another advantage of the polymer computer is that the unitary operations can go on in parallel, although the ability to address specific pairs of spins is lost.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [15] [15] S. Lloyd, 'A potentially realizable quantum computer', Science, 261 , 1569 (1993); 'Envisioning a quantum supercomputer', Science, 263 , 695 (1994). - source

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation
    § B. Gearbox quantum computer · p. 7

    One way to prepare an initially polarized state is to prepare a gear with attached single electron spins, polarize them at a reasonable temperature because of their greater magnetic moment, mesh them into contact with nuclear spins, and use known techniques such as the Overhauser effect or coherence transfer to transfer the electronic spin polarization to the nuclei.

    “One way of doing this would be to prepare a gear with attached single electron spins. Because of their much greater magnetic moment, these can be put in a polarized state at a reasonable temperature. When they are meshed into contact with the nuclear spins, one can use one of the known techniques (the Overhauser effect, coherence transfer [20]), for transferring the electronic spin polarization to”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [20] [20] C. P. Slichter, Principles of Magnetic Resonance (Third Edition, Springer-Verlag, 1992). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S1 (asserted), S2 (checkable), S3 (load-bearing).

    Interpretationcitation
    § IV. CONCLUSIONS · p. 15

    Other unique quantum phenomena like teleportation could also be investigated.

    “Other unique quantum phenomena like 'teleportation' [30] could also be investigated.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [30] [30] C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Perez, and W. Wooters, 'Teleport- ing an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels', Phys. Rev. Lett. 70 , 1895 (1993). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § B. Gearbox quantum computer · p. 8

    It may be difficult to make a magnetic polymer or a magnetic crystal sufficiently quiet, i.e., sufficiently immune from interaction with other, non-computational degrees of freedom.

    “Finally, there is a concern that it may be difficult to make a magnetic polymer or a magnetic crystal sufficiently 'quiet', i.e., sufficient immune from interaction with other, non-computational degrees of freedom.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § IV. CONCLUSIONS · p. 15

    Coppersmith has shown how both the essentially quantum-mechanical parts of Shor's algorithm and the conventional reversible part may be very efficiently designed in two-bit gates.

    “For the specific case of the Shor algorithm, Coppersmith [6] has very cleverly shown how both the essentially quantum-mechanical parts of his algorithm, and the 'conventional' reversible part, may be very efficiently designed in two-bit gates.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [6] [6] D. Coppersmith, 'An approximate Fourier transform useful in quantum factoring', preprint (1994); and unpublished. -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § B. Gearbox quantum computer · p. 7

    If more bits are needed for I/O, memory, etc., they can be added simply by adding more gears to the system.

    “It is obvious that if more bits are needed, for I/O, memory, etc., they can be added simply by adding more gears to the system.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § B. Gearbox quantum computer · p. 7

    There is presently no magnetometry of sufficient sensitivity to sense the state of a single proton spin, but mechanical detection of magnetic resonance within magnetic force microscopy will, according to Rugar and coworkers, be able to perform such detection in the foreseeable future.

    “There is presently no magnetometry of sufficient sensitivity to sense the state of a single proton spin; however, the mechanical detection of magnetic resonance within magnetic force microscopy will, according to Rugar and coworkers [19], be able to perform such detection in the foreseeable future.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [19] [19] D. Rugar, C. S. Yannoni and J. A. Sidles, 'Mechanical detection of magnetic resonance', Nature 360 , 563 (1992); O. Z¨ uger and D. Rugar, 'First images from a magnetic resonance force microscope', Appl. Phys. Lett. 63 , 2496 (1993). -

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § Abstract · p. 1

    Simpler versions of the gearbox computer could be used to do experiments on Einstein-Podolsky-Rosen states and related entangled quantum states.

    “Simpler versions of the gearbox computer could be used to do experiments on Einstein-Podolsky-Rosen states and related entangled quantum states.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation
    § IV. CONCLUSIONS · p. 15

    Shor's factoring algorithm gives hope that closely related problems such as graph isomorphisms might also have a rapid solution.

    “The main definite result for the time being is Shor's factoring algorithm, which gives the hope that closely related problems such as graph isomorphisms [28] might also have a rapid solution.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [28] [28] P. Shor, private communication. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim.

    Interpretationcitation
    § B. Gearbox quantum computer · p. 7

    It is expected that, as in Lloyd's work, suitable external fields can induce any arbitrary two-spin unitary operation, although no detailed protocol is worked out.

    “It is expected that, as in the work of Lloyd [15], it is possible with suitable external fields to induce any arbitrary two-spin unitary operation in Eq. (2.1), although I have not worked out a detailed protocol for this.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [15] [15] S. Lloyd, 'A potentially realizable quantum computer', Science, 261 , 1569 (1993); 'Envisioning a quantum supercomputer', Science, 263 , 695 (1994). - source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § B. Gearbox quantum computer · p. 6

    When the spins are not in contact, the spin Hamiltonian is zero and no time evolution occurs; during contact the spin Hamiltonian is non-zero, inducing a unitary transformation.

    “When the spins are not in contact, the Hamiltonian H spin of the spins is zero, and no time evolution occurs. During the time that the spins are in contact, H spin ( t ) will be non-zero, inducing, according to the well-known laws of quantum mechanics [18], the unitary transformation”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [18] [18] J. W. Negele and H. Orland, Quantum Many-Particle Systems , (Addison-Wesley, 1988), Eq. (2.11). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationreasoning
    § IV. CONCLUSIONS · p. 15

    By spatially separating the EPR pair and performing single-spin measurements on the two, one would observe the space-like non-locality unique to quantum mechanics and learn crucial information about dephasing times for pairs of spins.

    “By spatially separating this pair and performing single-spin measurements on the two, one would observe the space-like non-locality unique to quantum mechanics, and learn crucial information about dephasing times for pairs of spins.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § Abstract · p. 1

    Two-bit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins.

    “Two-bit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § B. Gearbox quantum computer · p. 7

    Since a swap is one of the simplest unitary operations, the state of any spin may be propagated arbitrarily far along an array of gears.

    “Note that since one of the simplest unitary operations is a swap, the state of any spin may be propagated arbitrarily far along an array of gears.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § A. Why making a quantum computer is very, very difficult · p. 4

    The subsystem isolation required for quantum computation will probably only be achievable if the computer elements are themselves of atomic or near-atomic dimensions, where the computational state is the quantum state of a single atom.

    “The kind of subsystem-isolation which quantum computation requires will probably only be achievable if the computer elements are themselves of atomic or near-atomic dimensions, where the computational state is the quantum state of a single atom.”
    Quote in the paperNot a substantial claim
27 of 27

  • The paper relies on this without support.

    Very long dephasing times for spins can be achieved in the gearbox by making the gear elements magnetically and electronically quiet.

    “By making the gear elements very 'quiet' magnetically and electronically, it may be hoped that a very long dephasing time for the spins may be achieved.”

  • The paper relies on this without support.

    Atomic spatial resolution and atomic contact between spins can be achieved in the gearbox.

    “The teeth of the left-hand gear are shown in the shape of atomic-force microscope (AFM) tips, suggestive of the fact that atomic spatial resolution will be necessary in the meshing of the two gears, in order that the two spins may be brought into atomic contact.”
2 of 2

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • A wide variety of physical systems-ranging from nearest neighbor Ising models to quantum electrodynamics and conformal field theories-are known to be computationally universal.

    “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 [57]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Dephasing time in seconds does not match years

    Minor

    The text gives the same dephasing time as 1010 seconds and 3 years. Ten billion seconds is about 317 years, not 3 years.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 11

  1. E. Bernstein, 1993, Proceedings of the 25th Annual ACM Symposium on the Theory of Computing
  2. P. Shor, 1994, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science
  3. D. R. Simon, Proceedings of the 35th Annual Symposium on the Foundations of Computer Science
  4. A. Abragham, 1961, The Principles of Nuclear Magnetism
  5. C. P. Slichter, 1992, Principles of Magnetic Resonance, 3rd ed.
  6. A. C. C. Yao, 1993, Proceedings of the 34th Annual Symposium on the Foundations of Computer Science
  7. J. Mathews, 1970, Mathematical Methods of Physics, 2nd ed.
  8. J. D. Talman, 1968, Special Functions, A Group Theoretic Approach
  9. A. Berthiaume, Proceedings of the Workshop on Physics and Computation
  10. A. Peres, 1993, Quantum Theory: Concepts and Methods
  11. R. M. Eisberg, 1974, Quantum Physics of Molecules, Solids, Nuclei, and Particles

Cited by

Sorted by citations

Authors
DiVincenzo, D. P.
Affiliations
Cambridge Scientific (United States) (US)
Corresponding author
DiVincenzo, David P.
ORCID
DiVincenzo, David P.
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
1995-02-01
Volume
51
Issue
2
Pages
1015-1022
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.51.1015
OpenAlex
W1993248778
Semantic Scholar
2c9a9dfadf7a6c1b05811a4052a15498fd9c1e9a
Cited by
1,069
References
23
Citations by year
2026: 16 · 2025: 39 · 2024: 37 · 2023: 19 · 2022: 25 · 2021: 33 · 2020: 29 · 2019: 29 · 2018: 21 · 2017: 13 · 2016: 22 · 2015: 31 · 2014: 34 · 2013: 30 · 2012: 29
FWCI
47.05
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
two-qubit gates, quantum computing, commutator algebra, Toffoli gate, magnetic resonance, nuclear spin, electron spin, atomic force microscopy, quantum phase coherence, Einstein-Podolsky-Rosen states, entangled states

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