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Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement
Open accessAbstract
Shor's powerful quantum algorithm for factoring represents a major challenge in quantum computation. Here, we implement a compiled version in a photonic system. For the first time, we demonstrate the core processes, coherent control, and resultant entangled states required in a full-scale implementation. These are necessary steps on the path towards scalable quantum computing. Our results highlight that the algorithm performance is not the same as that of the underlying quantum circuit and stress the importance of developing techniques for characterizing quantum algorithms.
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TL;DR
The paper experimentally implements compiled order-finding circuits for Shor’s algorithm factoring \(N=15\) with \(C=2\) and \(C=4\), using photonic qubits and measurement-induced nonlinearity. It reports near-ideal algorithm outputs but substantially lower circuit/process fidelities, highlighting that algorithm performance is not the same as underlying circuit performance.
Key findings
- The order-2 circuit gave logical probabilities \(\{P_{00},P_{10}\}=\{52,48\}\pm3\%\), and the order-4 circuit gave \(\{P_{000},P_{010},P_{100},P_{110}\}=\{27,23,24,27\}\pm2\%\); after classical processing, half the time the routines yield \(r=2\) or \(r=4\), giving factors 3 and 5.
- Argument-register state fidelities were \(F=99.9\pm0.3\%\) for order-2 and \(F=98.5\pm0.6\%\) for order-4, with linear entropies \(SL=100\pm1\%\) and \(SL=98.1\pm0.8\%\), respectively.
- Post-modular-exponentiation entangled states showed lower circuit performance: order-2 GHZ fidelity \(F_{\mathrm{GHZ}}=59\pm4\%\), \(SL=62\pm4\%\), witness \(W_{\mathrm{GHZ}}=-9\pm4\%\); order-4 Bell-product fidelity \(F=68\pm3\%\), \(SL=52\pm4\%\), with Bell-state tangles \(41\pm5\%\) and \(33\pm5\%\).
- Process fidelities for the order-4 two-qubit gates were \(F_p=85\%,89\%\), giving product circuit fidelity \(F_p=80\%\), versus an algorithm success rate of \(99.7\%\); order-2 decomposed three-qubit gate process fidelities were \(78\%,90\%\), with only a non-useful bound \(F_p\ge20\%\).
Methods
- Photonic linear-optics implementation with polarization-encoded qubits from parametric downconversion, using measurement-induced nonlinearity and nondeterministic gates with one-third success probability when fully prebiased; runs were flagged by fourfold coincidence events.
- Compiled circuits for \(N=15\): the order-2 circuit was implemented directly, while the order-4 circuit was further compiled to a pair of cnot gates; the paper used quantum state and process tomography, with convex-optimization tomography and Monte-Carlo error estimation.
- Analysis included measured density matrices, fidelities, linear entropies, entanglement witnesses/tangles, process \(\chi\)-matrices, and logical correlations between argument and function registers.
Limitations
- The paper states that current experiments are not scalable, that compiled algorithms do not directly imply scalability, and that direct implementation of the order-4 six-photon circuit is currently infeasible due to low six-photon rates and nondeterministic gates.
- It also states that full tomographic characterization becomes exponentially impractical for larger circuits, that order-2 process tomography would require at least 4096 measurements and was infeasible with their count rates, and that the three-qubit gate error bound is not useful.
- Evident from the text: this is a small-scale, compiled demonstration for \(N=15\), not a full Shor implementation; the measured entangled states partially overlap with ideal states and show environmental decoherence, and the circuit fidelities are much lower than the algorithm success rates.
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16 Not checked yet · 11 No basis given · 10 Own result · 5 Own interpretation
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“We measured relative nonclassical visibilities, V r ≡V meas /V ideal , of 98±2% and 85±6%.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“For the first time we demonstrate the core processes, coherent control, and resultant entangled states that are required in a full-scale implementation of Shor's algorithm.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Abstract
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“Here we implement a compiled version of Shor's algorithm in a photonic system using single photons and employing the non-linearity induced by measurement.”
- Also stated in: § Abstract · plain
“Here we implement a compiled version of Shor's algorithm, using photonic quantum-logic gates to realise the necessary processes, and verify the resulting entanglement via quantum state and process tomography [17,18].”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Abstract
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“we measured process fidelities of F p =85%, 89% for the two-qubit gates of the order-4 circuit.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“The results in Fig. 5 show there is a clear correlation between the argument and function registers, 59 to 83% and 67 to 87% for the order-2 and order-4 circuits, respectively.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“We have experimentally implemented every stage of a small-scale quantum algorithm.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Decomposing the three-qubit gate into a pair of two-qubit gates yields process fidelities F p =78%, 90% (again reflecting differing interferences of independent and dependent photons).”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Clearly this is significantly less than the algorithm success rate of 99.7%.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Consequently, by assuming that only these gates induce error, the order-4 circuit process fidelity is simply the product of the individual gate fidelities [32], F bcde p =F bd p F ce p =80%.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Again, these indicative values of circuit operation are significantly less than the algorithm success rates.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“We measure output probabilities of 25% to within error, as expected.After classical processing half the time the algorithm finds r=4, again yielding the factors 3 and 5.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“In the order-2 circuits the binary outputs of the algorithm are 00 or 10: the former represents the expected failure mode of this circuit, the latter a successful determination of r=2; failure and success should have equal probabilities, we measure them to be 50% to within error.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“The joint state of both registers after modular exponentiation indicates circuit performance; we find entangled states that partially overlap with the expected states, Fig. 4, indicating some environmental decoherence.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Thus the order-4 circuit, Fig. 1e), reduces to a pair of cnots, allowing us to implement the circuit in Fig. 1g).”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“We use a pair of compact optical gates [8,9,10], Fig 2c),e), each operating on a dependent pair of photons, resulting in measured visibilities for both of V r =98±2%.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“We present two different implementations of the order-finding routine at the heart of Shor's algorithm, characterising the algorithmic and circuit performances.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Measured fidelity with the ideal state, a tensor product of two Bell-states, is F =68±3%.The state is partially-mixed, SL=52±4%, and entangled, with tangles of the component Bell-States of 41±5% and 33±5%.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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“We use a linear-optical architecture where the required nonlinearity is induced by measurement; current experiments are not scalable, but there are clear paths to a fully scalable quantum architecture [19,20].”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Abstract
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“Directly encoding the order-4 finding circuit, Fig. 1e), requires six photons and at least one three-qubit and five two-qubit gates.This is currently infeasible: the best sixphoton rate to date [12] is 30 mHz, which would be reduced by six orders of magnitude using non-deterministic gates.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Thus half the time the algorithm yields r=2, which gives the factors, 3 and 5.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Shor's algorithm can factor a k-bit number using 72k 3 elementary quantum gates, e.g.factoring the smallest meaningful number, 15, requires 4608 gates operating on 21 qubits [13].”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Abstract
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“The fidelity of the four-qubit state with the ideal, Fig. 4b), is higher than that of the three-qubit state, Fig. 4a), chiefly because the latter requires nonclassical interference of photons produced by independent sources, which suffer higher distinguishability, lowering gate performance [29,30,31].”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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- Not checked yetcites [29]
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“However, since the qubits are at all times in a highly mixed state [15], and the dynamics can be fully modelled classically [16], neither the entanglement nor the coherent control at the core of Shor's algorithm can be implemented or verified.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Abstract
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“For all the circuits Fig. 1b)-g), the consecutive Hadamards in the top qubit of the argument-register cancel each other out (since h 2 =i): consequently both this qubit, and the gate(s) controlled by it, are redundant and need not be implemented experimentally.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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“Order-finding routines are a specific case of phase-estimation routines, which in turn underpin a wide variety of quantum algorithms, such as those in quantum chemistry [34].”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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- Not checked yetcites [34]
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“However, a bound can be obtained by chaining the gate errors, F p ≥20% [33].This is not a useful bound, c.f. the fidelity between an ideal cz and doing nothing at all of F p =25%!”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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- Not checked yetcites [33]
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“We use convex optimisation tomography [26] to estimate errors via Monte-Carlo simulation [6].”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > References
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“Application of Hadamard gates in the inverse QFT reset the |k qubits to 0, inhibiting any gates controlled by them”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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“The |k qubits do not become entangled to the function-register whereas the |a qubits are maximally-entangled to it-consequently after tracing out the function-register, the |a qubits are in a maximally-mixed state and any further gates acting on them are redundant.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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“After modular exponentiation, the circuit state is 2 n -1 x=0 |x |C x modN : for any two values x and y that differ by an integer, k number of orders, i.e. y-x=k 2 l , C y modN =C x modN , and the state after modular exponentiation becomes 2 n-l -1 k=0 2 l -1 a=0 |k2 l +a |C a modN .”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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“The remaining argumentregister qubits are maximally-entangled to the functionregister.Since the function-register output is not measured, these argument qubits are maximally-mixed, and the subsequent gates in the inverse QFT are therefore also redundant.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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“For larger circuits, full tomographic characterisation becomes exponentially impractical.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > FIG. 1: a)
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“Thus the inverse QFT in Ref. [14] was unnecessary: indeed, it is straightforward to show this is true for any order-2 l circuit.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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- Not checked yetcites [14] source
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“The final step of the inverse QFT is to swap the first and last qubits of the argument register which can be done after measurement.”
§ Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement > Additional Online Material.
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Only the two-qubit gates induce error in the order-4 circuit.
“by assuming that only these gates induce error”
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“Recently, a compiled version of Shor's algorithm has been demonstrated using four photonic qubits by Lu et al. (2007) and Lanyon et al. (2007).”
From Multiphoton entanglement and interferometry (2012) · cites this paper as Lanyon et al. (2007)
In the citing paper
“Recently there has also been substantial progress in investigations of other possible candidates for AQS, such as photonic systems [31, 32, 34-36], photonic and polariton systems [33, 38-42], artificial lattices in solid state systems [44].”
From Can one trust quantum simulators? (2012) · cites this paper as [31]
Citations
Cites
A scheme for efficient quantum computation with linear optics
Knill, E., Laflamme, R., & Milburn, G. J. (2001) · Nature · doi:10.1038/35051009 · Cited by 5,880
James, D. F. V., Kwiat, P. G., Munro, W. J., & White, A. G. (2001) · Physical Review A · doi:10.1103/physreva.64.052312 · Cited by 2,176
Self-citationExperimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance
Vandersypen, L. M. K., Steffen, M., Breyta, G., Yannoni, C. S., Sherwood, M. H., & Chuang, I. L. (2001) · Nature · doi:10.1038/414883a · Cited by 1,738
Simulated Quantum Computation of Molecular Energies
Aspuru-Guzik, A., Dutoi, A. D., Love, P. J., & Head-Gordon, M. (2005) · Science · doi:10.1126/science.1113479 · Cited by 1,418
Experimental one-way quantum computing
Walther, P., Resch, K. J., Rudolph, T., Schenck, E., Weinfurter, H., Vedral, V., Aspelmeyer, M., & Zeilinger, A. (2005) · Nature · doi:10.1038/nature03347 · Cited by 1,273
Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate
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Metadata
- Authors
- Lanyon, B. P., Weinhold, T. J., Langford, N. K., Barbieri, M., James, D. F. V., Gilchrist, A., & White, A. G.
- Affiliations
- The University of Queensland (AU)
- University of Toronto (CA)
- ORCID
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2007-12-19
- Volume
- 99
- Issue
- 25
- Pages
- 250505-250505
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1783862658
- Semantic Scholar
- 7699daaa2beac5d1aecea718fd4c5a433e639b4a
- Cited by
- 327
- References
- 27
- Citations by year
- 2026: 8 · 2025: 16 · 2024: 23 · 2023: 26 · 2022: 19 · 2021: 18 · 2020: 13 · 2019: 20 · 2018: 16 · 2017: 11 · 2016: 10 · 2015: 17 · 2014: 14 · 2013: 22 · 2012: 21
- FWCI
- 22.88
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Iris (Roma Tre University) (submittedVersion)
- Griffith Research Online (submittedVersion)
- arXiv (Cornell University)
- Funders
- Australian Research Council
- 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, photonic quantum computing, quantum entanglement, coherent control, quantum circuit compilation, scalable quantum computing
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