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Cited by 604Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyQuantum Mechanics and Applications

Experimental realization of a quantum algorithm

Open access
Chuang, Isaac L., Vandersypen, Lieven M. K., Zhou, Xinlan, Leung, Debbie W., Lloyd, Seth·1998·Nature
DOI 10.1038/30181
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Abstract

Quantum computers can in principle exploit quantum-mechanical effects to perform computations (such as factoring large numbers or searching an unsorted database) more rapidly than classical computers,,. But noise, loss of coherence, and manufacturing problems make constructing large-scale quantum computers difficult. Although ion traps and optical cavities offer promising experimental approaches,, no quantum algorithm has yet been implemented with these systems. Here we report the experimental realization of a quantum algorithm using a bulk nuclear magnetic resonance technique, in which the nuclear spins act as ‘quantum bits’. The nuclear spins are particularly suited to this role because of their natural isolation from the environment. Our simple quantum computer solves a purely mathematical problem in fewer steps than is possible classically, requiring fewer ‘function calls’ than a classical computer to determine the global properties of an unknown function.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper reports an NMR experimental realization of the simplest Deutsch–Jozsa quantum algorithm, using two nuclear spins in carbon-13-labeled chloroform to distinguish constant from balanced one-bit functions with one function call. The authors characterize the final two-qubit state and find small experimental errors, while noting that scaling NMR quantum computers beyond about 10 qubits will require new approaches.

Key findings

  • The simplest Deutsch–Jozsa algorithm was implemented on two qubits—the \(^1\)H and \(^{13}\)C nuclear spins in \(^{13}\)C-labeled chloroform—to determine whether a one-bit function is constant or balanced using only one function call; classically, deterministic solution requires up to \(2^{N-1}+1\) calls (2 calls for \(N=1\)).
  • The measured deviation density matrix showed normalized pure-state population varying from 0.998 to 1.019 (ideal 1), other elements smaller than 0.075 in magnitude, and relative error \(\epsilon\) between 8 and 12%.
  • Relaxation times were \(T_1 \approx 19\) and 25 s, and \(T_2 \approx 7\) and 0.3 s for proton and carbon, respectively; the experiment finished in about 7 ms.
  • Carbon signal-to-noise was \(\approx 35\), versus \(\approx 4300\) for proton.

Methods

  • Sample: 200 mM, 0.5 ml of \(^{13}\)C-labeled chloroform dissolved in d6-acetone at room temperature and standard pressure; the \(O(10^{18})\) molecules were treated as independent single quantum computers. Experiments used an 11.7 Tesla Oxford Instruments magnet and a Varian UNITY Inova spectrometer with a triple-resonance probe.
  • Algorithm implementation: RF pulses and scalar spin-spin coupling \(J \approx 215\) Hz; typical pulse lengths were 10–15 µs, and \(\tau/2 = 1/(4J) \approx 1.163\) ms. Readout used a phase-sensitive pick-up coil and spectra; full deviation density matrix was obtained by quantum state tomography from 9 experiments.
  • Analysis: temporal averaging with three experiments cyclically permuted populations to extract the effective pure \(|00\rangle\) signal; the paper notes that although this method evaluates \(f(x)\) three times, the algorithm also works with a thermal input state and only one experiment is needed.

Limitations

  • Stated: the most important error source was RF field inhomogeneity and pulse length calibration imperfections; the second most important was low carbon signal-to-noise. Smaller contributions came from incomplete relaxation between experiments, carrier frequency offsets, and numerical errors in data analysis.
  • Stated: NMR quantum computers larger than about 10 qubits will require creative new approaches because signal strength decays exponentially with qubit number; coherence times typically decrease for larger molecules while average logic gate duration increases.
  • Evident from the text: the demonstration is limited to a 2-qubit implementation of the simplest Deutsch–Jozsa case, so it does not itself demonstrate scaling to larger algorithms or machines.

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

Integrity signals

Registry

Crossref · 10.1038/30181

Cites

24 of 24 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 26 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § Experimental realization of a quantum algorithm · p. 1

    The quantum algorithm was experimentally implemented using the nuclear spins of the 1H and 13C atoms in a carbon-13 labeled chloroform molecule as the input and work qubits.

    “Experimentally, this quantum algorithm was implemented using the nuclear spins of the 1 Hand 13 Catoms in a carbon-13 labeled chloroform molecule (CHCl 3 ) as the input and work quantum bits ('qubits').”
    Quote in the paperSubstantial claim
  • Findingnone given
    § Experimental realization of a quantum algorithm · p. 1

    Their simple quantum computer can determine global properties of an unknown function using fewer function calls than is possible using a classical computer.

    “In particular, our simple quantum computer can determine global properties of an unknown function using fewer function 'calls' than is possible using a classical computer.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: To determine with certainty whether a function is constant or balanced on a deterministic classical computer requires up to 2^{N-1}+1 function calls.
  • Findingown data
    § Experimental realization of a quantum algorithm · p. 3

    The relative error ǫ on the experimental pure-state output density matrix ρ_exp, defined as ǫ = ||ρ_exp - ρ_theory|| / ||ρ_theory||, varied between 8 and 12%.

    “The relative error ǫ on the experimental pure-state output density matrix ρ exp , defined as ǫ = ‖ ρ exp -ρ theory ‖ / ‖ ρ theory ‖ , varied between 8 and 12%.”
    Quote in the paperNot a substantial claim
  • Findingown data
    § Experimental realization of a quantum algorithm · p. 3

    The other deviation density matrix elements (ideally 0) were smaller than 0.075 in magnitude.

    “The other deviation density matrix elements (ideally 0), were smaller than 0 . 075 in magnitude.”
    Quote in the paperNot a substantial claim
  • Findingreasoning
    § Experimental realization of a quantum algorithm · p. 3

    The second most important contribution to errors is the low carbon signal-to-noise ratio, signal peak height / RMS noise ≈ 35, versus ≈ 4300 for proton.

    “The second most important contribution to errors is the low carbon signal-to-noise ratio, signal peak height / RMS noise ≈ 35, versus ≈ 4300 for proton.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 12
  • Classified as probably not a substantial claim. 6
  • The quote only partly states this claim. 1

  • May combine two claims.

    Findingown data
    § Experimental realization of a quantum algorithm · p. 2

    These relaxation times were much longer than required for the experiment, which finished in about 7 milliseconds.

    “these were much longer than required for our experiment, which finished in about 7 milliseconds.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Experimental realization of a quantum algorithm · p. 2

    These results unambiguously demonstrate the complete proper functioning of the quantum algorithm and provide data for the error analysis.

    “These results unambiguously demonstrate the complete proper functioning of the quantum algorithm, and provide data for the error analysis described below.”
    Quote in the paperSubstantial claim

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

    Findingcitation
    § Experimental realization of a quantum algorithm · p. 1

    The version of the D-J algorithm realized is the simplest possible case, where f is a function from one bit to one bit, which is also the simplest instance of Simon's algorithm.

    “The D-J algorithm is well illustrated by its simplest possible case, when f is a function from one bit to one bit; this is the version that we have realized (it is also the simplest instance of Simon's algorithm 7 ).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [7] Simon, D., 'On the power of quantum computation,' Proc. 35 th Ann. Symp. on Found. of Computer Science (IEEE Comp. Soc. Press, Los Alamitos, CA, 1994) 116-123. source

      Not checked in the free run.

  • May combine two claims.

    Findingown data
    § Experimental realization of a quantum algorithm · p. 1

    The authors experimentally realized a quantum algorithm using NMR, solving a purely mathematical problem in fewer steps than is possible classically.

    “Here, we report on the experimental realization of a quantum algorithm using NMR, to solve a purely mathematical problem in fewer steps than is possible classically.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: To determine with certainty whether a function is constant or balanced on a deterministic classical computer requires up to 2^{N-1}+1 function calls.

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

    Findingnone given
    § Experimental realization of a quantum algorithm · p. 3

    Smaller contributions to errors came from incomplete relaxation between subsequent experiments, carrier frequency offsets, and numerical errors in the data analysis.

    “Smaller contributions to errors came from incomplete relaxation between subsequent experiments, carrier frequency offsets, and numerical errors in the data analysis.”
    Quote in the paperNot a substantial claim

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

    Premisecitation
    § Experimental realization of a quantum algorithm · p. 1

    Noise, decoherence, and manufacturing problems make constructing large-scale quantum computers difficult.

    “Noise, decoherence, and manufacturing problems make constructing large-scale quantum computers difficult 9-13 .”
    Quote in the paperNot a substantial claim0 of 5 sources checked
    • cites [9] Unruh, W. G., 'Maintaining coherence in quantum computers,' Phys. Rev. A 51 , 2, 992-997 (1995). source

      Not checked in the free run.

    • cites [10] Chuang, I. L., Laflamme, R., Shor, P. and Zurek, W. H., 'Quantum computers, factoring, and decoherence,' Science 270 , 5242, 1633-1635 (1995). source

      Not checked in the free run.

    • cites [11] Landauer, R., 'Dissipation and Noise Immunity in Computation and Communication,' Nature , Vol. 335 , 779-784 (1988). source

      Not checked in the free run.

    • cites [12] Landauer, R., 'Is Quantum Mechanics Useful?' Phil. Trans. R. Soc. Lond. A , 335 , 367-376 (1995).

      Not checked in the free run.

    • cites [13] Palma, G.M., Suominen, K.-A. and Ekert, A.K., 'Quantum Computers and Dissipation,' Proc. R. Soc. Lond., A , 452 , 567-584 (1996). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Experimental realization of a quantum algorithm · p. 1

    A quantum computer can in principle exploit coherent quantum interference and entanglement to perform computations such as factoring large numbers or searching an unsorted database more rapidly than classical computers.

    “In principle, it can exploit coherent quantum interference and entanglement to perform computations, such as factoring large numbers or searching an unsorted database, more rapidly than classical computers 1,2,6-8 .”
    Quote in the paperSubstantial claim0 of 5 sources checked
    • cites [1] Deutsch, D. 'Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer,' Proc. R. Soc. Lond., A , 400 , 97-117 (1985). source

      Not checked in the free run.

    • cites [2] Shor, P., 'Algorithms for quantum computation: discrete logarithms and factoring,' Proc. 35 th Ann. Symp. on Found. of Computer Science (IEEE Comp. Soc. Press, Los Alomitos, CA, 1994) 124-134 . source

      Not checked in the free run.

    • cites [6] Deutsch, D. and Jozsa, R., 'Rapid solution of problems by quantum computation,' Proc. R. Soc. Lond., A 439 , 553-558 (1992). source

      Not checked in the free run.

    • cites [7] Simon, D., 'On the power of quantum computation,' Proc. 35 th Ann. Symp. on Found. of Computer Science (IEEE Comp. Soc. Press, Los Alamitos, CA, 1994) 116-123. source

      Not checked in the free run.

    • cites [8] Grover, L. K., 'Quantum computers can search arbitrarily large databases by a single query,' Phys. Rev. Lett. 79 , 23, 4709-4012 (1997). source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § Experimental realization of a quantum algorithm · p. 1

    Ion traps and optical cavities offer promising experimental approaches, but no quantum algorithm has yet been implemented with those systems.

    “Ion traps and optical cavities offer promising experimental approaches 14,15 , but no quantum algorithm has yet been implemented with those systems.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [14] Monroe, C., Meekhof, D.M., King, B.E., Itano, W.M. and Wineland, D.J., 'Demonstration of a fundamental quantum logic gate,' Phys. Rev. Lett. 75 , 25, 4714-4717 (1995). source

      Not checked in the free run.

    • cites [15] Turchette, Q. A., Hood, C.J., Lange, W., Mabuchi, H. and Kimble, H.J., 'Measurement of conditional phase shifts for quantum logic,' Phys. Rev. Lett. 75 , 25, 4710-4713 (1995). source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § Experimental realization of a quantum algorithm · p. 1

    Nuclear spins are particularly good quantum bits because of their natural isolation from the environment, and their use for quantum computation is possible by applying NMR techniques in an unconventional manner.

    “because of their natural isolation from the environment, nuclear spins are particularly good 'quantum bits' 16 , and their use for quantum computation is possible by applying nuclear magnetic resonance (NMR) techniques in an unconventional manner 17-19 .”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [16] Lloyd, S., 'A potentially realizable quantum computer,' Science 261 , 5128, 1569-1571 (1993). source

      Not checked in the free run.

    • cites [17] Gershenfeld, N. and Chuang, I. L., 'Bulk spin-resonance quantum computation,' Science 275 , 5298, 350-356 (1997). source

      Not checked in the free run.

    • cites [18] Cory, D. G., Price, M. D., Fahmy A. F. and Havel, T. F., 'Nuclear magnetic resonance spectroscopy: an experimentally accessible paradigm for quantum computing,' Physica D, in print; LANL E-print quant-ph/9709001.gov (1997).

      Not checked in the free run.

    • cites [19] Cory, D. G., Fahmy, A. F., and Havel, T. F., 'Ensemble quantum computing by NMR spectroscopy,' Proc. Nat. Acad. Sci. 94 , 1634-1639 (1997). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationnone given
    § Experimental realization of a quantum algorithm · p. 2

    The algorithm works equally well if the input qubit is initially |1>; when the work qubit is initially |1>, it fails and cannot distinguish constant from balanced functions, but this does not interfere with other computers which have worked.

    “Although step (T0) stipulates an input pure state | 00 〉 , the algorithm works equally well if the input qubit is initially | 1 〉 ; furthermore, when the work qubit is initially | 1 〉 , it fails, and cannot distinguish constant from balanced functions, but this does not interfere with other computers which have worked.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Premisecitation
    § Experimental realization of a quantum algorithm · p. 1

    The reduced Hamiltonian for the 2-spin system is to an excellent approximation given by the equation with free precession terms and a scalar spin-spin coupling of J ≈ 215 Hz.

    “The reduced Hamiltonian for this 2-spin system is to an excellent approximation given by (¯ h = 1) 21”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [21] Slichter, C. P., Principles of Magnetic Resonance. (Springer, Berlin, 1990).

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § Experimental realization of a quantum algorithm · p. 3

    The carbon signal was much weaker because the carbon gyromagnetic ratio is 4 times smaller, and the carbon receiver coil is mounted more remotely from the sample.

    “The carbon signal was much weaker because the carbon gyromagnetic ratio is 4 times smaller, and the carbon receiver coil is mounted more remotely from the sample.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § Experimental realization of a quantum algorithm · p. 2

    Thus, a thermal state is a good input for this algorithm, and only one experiment needs to be performed.

    “Thus, a thermal state is a good input for this algorithm, and only one experiment needs to be performed.”
    Quote in the paperSubstantial claim

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

    Interpretationreasoning
    § Experimental realization of a quantum algorithm · p. 3

    Optical pumping and other cooling techniques can be used to pre-polarize the sample to increase the output signal amplitude, since Z ≈ 1 at low temperatures.

    “Optical pumping and other cooling techniques can also be used to pre-polarize the sample to increase the output signal amplitude, since Z ≈ 1 at low temperatures.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § Experimental realization of a quantum algorithm · p. 2

    The temporal averaging technique avoids the technical difficulties of detecting the signal from a single nuclear spin and allows a room-temperature sample, which produces an easily detectable signal, to be used for quantum computation.

    “This technique avoids the technical difficulties of detecting the signal from a single nuclear spin, and allows a sample at room temperature, which produces an easily detectable signal, to be used for quantum computation.”
    Quote in the paperSubstantial claim

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

    Interpretationreasoning
    § Experimental realization of a quantum algorithm · p. 2

    Although the temporal averaging method requires f(x) to be evaluated 3 times, this is actually not necessary.

    “Note that while this method requires f ( x ) to be evaluated 3 times, it is actually not necessary.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Study design and conductcitation
    § Experimental realization of a quantum algorithm · p. 2

    The longitudinal and transverse relaxation time constants T1 and T2 were measured using standard inversion-recovery and Carr-Purcell-Meiboom-Gill pulse sequences, giving T1 ≈ 19 and 25 seconds, and T2 ≈ 7 and 0.3 seconds, respectively, for proton and carbon.

    “The longitudinal and transverse relaxation time constants T 1 and T 2 were measured using standard inversion-recovery and Carr-Purcell-MeiboomGill pulse sequences 23 , giving T 1 ≈ 19 and 25 seconds, and T 2 ≈ 7 and 0 . 3 seconds, respectively, for proton and carbon; these were much longer than required for our experiment, which finished in about 7 milliseconds.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [23] Ernst, R. R., Bodenhausen, G. and Wokaun, A., Principles of Nuclear Magnetic Resonance in One and Two Dimensions. (Oxford University Press, Oxford, 1994). source

      Not checked in the free run.

17 of 17

  • The paper relies on this without support.

    To determine with certainty whether a function is constant or balanced on a deterministic classical computer requires up to 2^{N-1}+1 function calls.

    “To determine with certainty whether a function is constant or balanced on a deterministic classical computer, requires up to 2 N -1 +1 function calls: even if one has looked at half of the inputs and found f ( x ) = 0 for each, one still can't conclude with certainty that the function is constant.”

  • The paper relies on this without support.

    Coherence times typically decrease for larger molecules, while the average logic gate duration increases.

    “Furthermore, coherence times typically decrease for larger molecules, while the average logic gate duration increases.”
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?

  • Stokes parameters directly yield the polarization density matrix for an ensemble of identically prepared single photons.

    “directly yield the polarization density matrix for an ensemble of identicallyprepared single photons [27]”

    From Nonmaximally Entangled States: Production, Characterization, and Utilization (1999) · cites this paper as [27]

    The discussion on p. 3 describes reconstructing a two-qubit nuclear-spin density matrix from proton and carbon spectral lines. Searches for Stokes parameters and single-photon polarization found no passage addressing whether Stokes parameters yield a polarization density matrix. (LLM, conf=0.77)

    Experimental — accuracy not yet measured

  • Prototype quantum computers that store and process information on individual atoms have already been demonstrated.

    “Although an Avogadro scale computer that can act on 10 23 bits might seem implausible, prototype quantum computers that store and process information on individual atoms have already been demonstrated 64 -65 , 76 -80 .”

    From Ultimate physical limits to computation (2000) · cites this paper as [78]

  • Recently constructed quantum microcomputers use one quantum degree of freedom for each bit and operate at the Heisenberg limit Δt = π hbar/2ΔE for the time needed to flip a bit.

    “The laws of physics do not require redundancy to perform logical operations: recently constructed quantum microcomputers use one quantum degree of freedom for each bit and operate at the Heisenberg limit ∆ t = π ¯ h/ 2∆ E for the time needed to flip a bit 64 -65 , 76 -80 .”

    From Ultimate physical limits to computation (2000) · cites this paper as [78]

  • Contemporary quantum microcomputers constructed using nuclear magnetic resonance operate at the limits of speed and memory space described above.

    “In fact, contemporary quantum 'microcomputers' such as those constructed using nuclear magnetic resonance 76 -80 do indeed operate at the limits of speed and memory space described above.”

    From Ultimate physical limits to computation (2000) · cites this paper as [78]

  • NMR quantum computers have previously been used to demonstrate quantum algorithms to solve the two bit Deutsch problem.

    “NMR quantum computers have previously been used to demonstrate quantum algorithms to solve the two bit Deutsch problem. 7, 8”

    From Implementation of a quantum search algorithm on a quantum computer (1998) · cites this paper as [8]

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. Landauer, R. Is quantum mechanics useful? Phil. Trans. R. Soc. Lond. A 335, 367–376 (1995).
  2. LANL E-print quant-ph/9709001.gov, 1997). · arXiv
  3. Cleve, R., Ekert, A., Macchiavello, C. & Mosca, M. Proc. R. Soc. Lond. A 454, 339–354 (1998). LANL E-prin quant-ph/9708016. · arXiv
  4. Jones, T. F. & Mosca, M. Implementation of a quantum algorithm to solve deutsch's problem on a nuclear magnetic resonance quantum computer. J. Chem. Phys.(in the press;

Cited by

Sorted by citations

Authors
Chuang, I. L., Vandersypen, L. M. K., Zhou, X., Leung, D. W., & Lloyd, S.
Affiliations
  • IBM Research - Almaden (US)
  • Stanford University (US)
  • Massachusetts Institute of Technology (US)
Corresponding author
Chuang, Isaac L.
ORCID
Venue
Nature
Publisher
Springer Science and Business Media LLC
Published
1998-05
Volume
393
Issue
6681
Pages
143-146
ISSN
  • 0028-0836 (ISSN-L)
  • 1476-4687
DOI
10.1038/30181
OpenAlex
W1784120331
Semantic Scholar
40947c615554c5aec882f4204615015b83981d75
Cited by
604
References
24
Citations by year
2026: 4 · 2025: 12 · 2024: 13 · 2023: 12 · 2022: 14 · 2021: 17 · 2020: 8 · 2019: 7 · 2018: 12 · 2017: 11 · 2016: 13 · 2015: 11 · 2014: 6 · 2013: 6 · 2012: 19
FWCI
72.21
Type
journal article
Language
en
Open access
Licence
www.springer.com/tdm (tdm)
Locations
  • Nature (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • RePEc: Research Papers in Economics (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 › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
quantum algorithms, quantum computing, nuclear magnetic resonance, nuclear spin, qubits, Grover's algorithm, unstructured search, quantum parallelism

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