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Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance
Open accessAbstract
The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes. Quantum computers, however, could factor integers in only polynomial time, using Shor's quantum factoring algorithm. Although important for the study of quantum computers, experimental demonstration of this algorithm has proved elusive. Here we report an implementation of the simplest instance of Shor's algorithm: factorization of N = 15 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits, which can be manipulated with room temperature liquid-state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to systems containing many quantum bits, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects in our system.
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TL;DR
The paper reports an NMR implementation of Shor’s quantum factoring algorithm, factoring N = 15 into 3 and 5 using seven spin-1/2 nuclei as qubits. It also presents a parameter-free decoherence model that reproduces the main non-ideal features of the measured spectra.
Key findings
- Shor’s algorithm was experimentally executed for N = 15 with two parameter choices: a = 11 (“easy” case) and a = 7 (“difficult” case). The output spectra gave period r = 2 for a = 11 and r = 4 for a = 7, yielding gcd(11^2/2 ± 1, 15) = 3, 5 and gcd(7^4/2 ± 1, 15) = 3, 5, respectively.
- The quantum computer used seven spin-1/2 nuclei: five ^19F and two ^13C, manipulated by room-temperature liquid-state NMR. The difficult case used a sequence of ∼300 spin-selective RF pulses, with the complete Shor sequence lasting up to ∼720 ms.
- A parameter-free decoherence model, using measured T1 and T2 values and treating generalized amplitude damping and phase damping sequentially, agreed well with the large non-idealities in the data. The paper states this is the first NMR quantum computation experiment for which decoherence is the dominant source of errors.
- The authors state that scalability to many qubit systems is not implied by this work; the significance lies in demonstrating experimental and theoretical techniques for precise control and modelling of complex quantum computers.
Methods
- Experiments were performed at IBM Almaden with an 11.7 T (500 MHz) magnet, a custom-modified four-channel Varian Unity INOVA spectrometer, and a Nalorac HFX probe. The molecule was a custom-synthesized perfluorobutadienyl iron complex with the inner two carbons ^13C-labeled, at 0.88 ± 0.04 mole % in diethyl ether-d10, with all experiments at 30 °C.
- The seven-qubit effective pure state was created by a two-stage temporal averaging scheme summing 4 × 9 = 36 experiments, with state preparation sequences ∼200 ms long. The algorithm was implemented with spin-selective Hermite-180 and Gaussian-90 pulses, shaped in 256 steps, and read out by NMR spectroscopy of the first three qubits.
- The modular exponentiation and inverse QFT were implemented with a simplified quantum circuit for N = 15, including controlled additions and swaps, followed by spectral analysis to determine the period and factors.
Limitations
- The paper states that scalability of this NMR approach to many qubit systems is not implied by the present work.
- The decoherence model assumes independent stochastic relaxation with correlation times ≪ 1/ω_i, uses phenomenological Bloch equations with only T1 and T2 per spin, and treats non-commuting processes as commuting/sequential; the authors note remaining discrepancies due to model approximations and experimental imperfections such as RF inhomogeneity, imperfect calibrations, incomplete field drift compensation, and incomplete unwinding of coupled evolution during RF pulses.
- The experiment used an ensemble of independent quantum computers, so measurement gives the bitwise average value of 8c/r rather than a sample of 8c/r; this was sufficient for the present experiment, but for larger N a continued fractions algorithm would need to be performed on the quantum computer, requiring additional qubits.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Here we report an implementation of the simplest instance of Shor's algorithm: factorization of N = 1 5 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits [11, 12], which can be manipulated with room temperature liquid state nuclear magnetic resonance techniques.”
- Also stated in: § Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance · p. 5 · boosted
“Thus, even after the long and complex pulse sequence of the difficult case, the experimental data conclusively indicate the successful execution of Shor's algorithm to factor 15.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Also stated in: § Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance · p. 5 · boosted
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The predictive value of the model was further confirmed via independent test experiments.”
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: Each spin experiences independent stochastic relaxation with correlation time scales much less than 1/ω_i.
- The authors state this as an assumption: The non-commuting GAD, PD, and ideal unitary evolution processes can be treated as commuting transformations so that simultaneous processes can be modeled sequentially.
In the paper
“This is the first NMR quantum computation experiment for which decoherence is the dominant source of errors [8]; the demands of Shor's algorithm clearly push the limits of the current molecule, despite its exceptional properties.”
What the sources say
- Not checked yetcites [8] Jones, J. A. NMR quantum computation. Prog. NMR Spectrosc. 38, 325–360 (2001). source
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesNone of the sources cited for this claim have been checked yet.
In the paper
“We converted ρ th into a 7-spin effective pure state [11, 12] ρ 1 via temporal averaging [9] (step 0); ρ 1 constitutes a suitable initial state for Shor's factoring algorithm since it generates the same signal as | ψ 1 〉 , up to a proportionality constant [11, 12].”
What the sources say
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIn the paper
“From r , at least one factor of N is given by the greatest common denominator (gcd) of a r/ 2 ± 1 and N (with probability greater than 1 / 2); the gcd can be computed efficiently using Euclid's algorithm on a classical computer [2] .”
What the sources say
- Not checked yetcites [2] Koblitz, N. A Course in Number Theory and Cryptography (Springer, New York, 1994). source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“At the same time, the good agreement between the measured and simulated spectra suggests that the degree of unitary control in the experiment was very high, which bodes well for related proposed implementations of quantum computers [28, 29].”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: Each spin experiences independent stochastic relaxation with correlation time scales much less than 1/ω_i.
- The authors state this as an assumption: The non-commuting GAD, PD, and ideal unitary evolution processes can be treated as commuting transformations so that simultaneous processes can be modeled sequentially.
None of the sources cited for this claim have been checked yet.
In the paper
“The two main components of the algorithm, modular exponentiation (computation of a x mod N ) and the inverse quantum Fourier transform (QFT) take only O ( l 3 ) operations [4, 5, 6].”
What the sources say
- Not checked yetcites [4] Shor, P. in Proc. 35th Annu. Symp. on the Foundations of Computer Science (ed. Goldwasser, S.) 124–134 (IEEE Computer Society Press, Los Alamitos, California, 1994). source
- Not checked yetcites [5] Shor, P. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997). source
- Not checked yetcites [6] Ekert, A. & Jozsa, R. Quantum computation and Shor's factoring algorithm. Rev. Mod. Phys. 68(3), 733–753 (1996). source
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- The quote only partly states this claim. 1
- It may state two separate claims as one. 11
- Probably not a substantial claim. 1
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Nevertheless, there are obvious discrepancies between the measured and ideal spectra, most notably for the difficult case.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe quote only partly states this claim.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The periodicity in the amplitude of | y 〉 is now 2, so r = 8 / 2 = 4 and gcd(7 4 / 2 ± 1 , 15) = 3 , 5. Thus, even after the long and complex pulse sequence of the difficult case, the experimental data conclusively indicate the successful execution of Shor's algorithm to factor 15.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“While some discrepancies between the data and the simulations remain (due to the approximations in the model as well as due to experimental imperfections such as RF inhomogeneity, imperfect calibrations, incomplete field drift compensation and incomplete unwinding of coupled evolution during the RF pulses [15]), the model agrees well with the large non-idealities of the data.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: Each spin experiences independent stochastic relaxation with correlation time scales much less than 1/ω_i.
- The authors state this as an assumption: The non-commuting GAD, PD, and ideal unitary evolution processes can be treated as commuting transformations so that simultaneous processes can be modeled sequentially.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The periodicity in the amplitude of | y 〉 is thus 4, so r = 2 n / 4 = 2 and we find that gcd(11 2 / 2 ± 1 , 15) = 3 , 5. The prime factors thus unambiguously derive from the output spectra.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“All seven spins in this molecule are remarkably well separated in frequency ω i / 2 π , and interact pairwise via the J -coupling, described by H J = ∑ i<j 2 π ¯ hJ ij I zi I zj [17].”
What the sources say
- Not checked yetcites [17] Schack, R. & Caves, C. M. Classical model for bulk-ensemble NMR quantum computation. Phys. Rev. A 60, 4354–4362 (1999). source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Using this simple model, the 7spin simulation of the complete Shor pulse sequence, including 36 temporal averaging sequences, required only a few minutes on an IBM quad power 3-II processor machine.”
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
- The authors state this as an assumption: Each spin experiences independent stochastic relaxation with correlation time scales much less than 1/ω_i.
- The authors state this as an assumption: The non-commuting GAD, PD, and ideal unitary evolution processes can be treated as commuting transformations so that simultaneous processes can be modeled sequentially.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Even for such a small N , quantum factorization poses at present a significant experimental challenge: it requires coherent control over seven quantum bits (qubits) in the course of a long sequence of controlled interactions, even after maximal reduction of the quantum circuit; including the state initialization, interactions between almost all pairs of qubits are needed.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“Shor's factoring algorithm works by utilizing a quantum computer to quickly determine the period of the function f ( x ) = a x mod N (the remainder of a x divided by N ), where a is a randomly chosen small number with no factors in common with N ; from this period, number-theoretic techniques can be employed to factor N with high probability [4].”
What the sources say
- Not checked yetcites [4] Shor, P. in Proc. 35th Annu. Symp. on the Foundations of Computer Science (ed. Goldwasser, S.) 124–134 (IEEE Computer Society Press, Los Alamitos, California, 1994). source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“The simplest meaningful instance of Shor's algorithm is factorization of N = 15 [7] - the algorithm fails for N even or a prime power.”
What the sources say
- Not checked yetcites [7] Beckman, D., Chari, A. N., Devabhaktuni, S. & Preskill, J. Efficient networks for quantum factoring. Phys. Rev. A 54, 1034–1063 (1996). source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“A full description of relaxation for the seven coupled spins involves almost 4 7 × 4 7 degrees of freedom and requires knowledge of physical properties of the molecule which are not available [23, 24].”
What the sources say
- Not checked yetcites [23] Kraus, K. States, Effects, and Operations: Fundamental Notions of Quantum Theory (Springer, Berlin, 1983). source
- Not checked yetcites [24] Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“Classically, in contrast, prime factorization takes O (2 l 1 / 3 ) operations [1], which quickly becomes intractable as l increases.”
What the sources say
- Not checked yetcites [1] Knuth, D. E. The Art of Computer Programming Vol. 2, Seminumerical Algorithms (Addison-Wesley, Reading, Massachusetts, 1998).
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“Thus, the first register can be as small as two qubits ( n = 2); however, three qubits ( n = 3) allow for the possibility of detecting more periods and thus constitutes a more stringent test of the modular exponentiation and QFT [15].”
What the sources say
- Not checked yetcites [15] Freeman, R. Spin Choreography (Spektrum, Oxford, 1997).
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“In the experiment, an ensemble of independent quantum computers rather than a single quantum computer was used, so the measurement gives the bitwise average value of 8 c/r , instead of a sample of 8 c/r . This is sufficient to determine r in the present experiment, but for larger N a continued fractions algorithm will need to be performed on the quantum computer [11] requiring additional qubits.”
What the sources say
- Not checked yetcites [11] Gershenfeld, N. & Chuang, I. L. Bulk spin-resonance quantum computation. Science 275, 350–356 (1997). source
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The authors state this as an assumption.
Each spin experiences independent stochastic relaxation with correlation time scales much less than 1/ω_i.
“In order to get a first estimate of the impact of decoherence during the factoring pulse sequence we assume that each spin experiences independent stochastic relaxation with correlation time scales ≪ 1 /ω i .”
The authors state this as an assumption.
The non-commuting GAD, PD, and ideal unitary evolution processes can be treated as commuting transformations so that simultaneous processes can be modeled sequentially.
“However, GAD does not commute with e -i H t , and PD and GAD do not commute with the ideal unitary evolution during RF pulses. Nevertheless, we have treated these as commuting transformations, such that all of the processes which occur simultaneously can be modeled sequentially.”
The paper relies on this without support.
An effective pure ground state of all seven spins had never been created before.
“To create an effective pure ground state of all seven spins, which has never been done before, we used a two-stage extension of the scheme of Ref. [9], necessary because ω 13 C is very different from ω 19 F .”
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The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“It is notable that in these optical experiments during the computation genuine multi-particle entanglement and multi-path interference were observed, which did not appear in the previous implementations using nuclear magnetic resonance (Vandersypen et al. 2001).”
From Multiphoton entanglement and interferometry (2012) · cites this paper as Vandersypen et al. 2001
Experimental — accuracy not yet measuredIn the citing paper
“In the only demonstration to date, a compiled set of gate operations were implemented in a liquid NMR architecture [14].”
From Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement (2007) · cites this paper as [14]
In the citing paper
“Thus the inverse QFT in Ref. [14] was unnecessary: indeed, it is straightforward to show this is true for any order-2 l circuit.”
From Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement (2007) · cites this paper as [14]
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Research works printed but not linked: 2
- Schulman, L. & Vazirani, U. in Proc. 31st ACM Symp. on Theory of Computing 322–329 (Association for Computing Machinery, New York, 1999).
- Freeman, R. Spin Choreography (Spektrum, Oxford, 1997).
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Microtubules as a Quantum Hopfield Network
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Solid-State Silicon NMR Quantum Computer
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NMR Implementation of a Parallel Search Algorithm
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Metadata
- Authors
- Vandersypen, L. M. K., Steffen, M., Breyta, G., Yannoni, C. S., Sherwood, M. H., & Chuang, I. L.
- Affiliations
- IBM Research - Almaden (US)
- Venue
- Nature
- Published
- 2001-12
- Volume
- 414
- Issue
- 6866
- Pages
- 883-887
- ISSN
- 0028-0836 (ISSN-L)
- 1476-4687
- OpenAlex
- W2159226773
- Semantic Scholar
- eec83aa6e6746cab27ca7b5dd7e9535646ee65a2
- Cited by
- 1,738
- References
- 34
- Citations by year
- 2026: 37 · 2025: 41 · 2024: 66 · 2023: 97 · 2022: 82 · 2021: 75 · 2020: 58 · 2019: 96 · 2018: 41 · 2017: 79 · 2016: 65 · 2015: 55 · 2014: 60 · 2013: 56 · 2012: 74
- FWCI
- 83.11
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Nature (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (acceptedVersion)
- RePEc: Research Papers in Economics (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 › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- Shor's algorithm, quantum factoring, integer factorization, nuclear magnetic resonance, quantum computing, spin-1/2 nuclei, decoherence, qubits
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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