DOCSLIB.ORG

Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes CORE Metadata, citation and similar papers at core.ac.uk Provided by DSpace@MIT PHYSICAL REVIEW X 6, 031039 (2016) Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes Theodore J. Yoder, Ryuji Takagi, and Isaac L. Chuang Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 12 March 2016; published 13 September 2016) It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here, we demonstrate precisely the existence of such gates. In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching. DOI: 10.1103/PhysRevX.6.031039 Subject Areas: Quantum Information I. INTRODUCTION they have important, and perhaps several important, trans- versal gates. One of the crucial concepts in error-correcting codes is Unfortunately, it is a well-known theorem [2–4] that that of logical circuits—a set of circuits fC g that give the i there is no quantum code with a universal set of transversal ability to carry out a set of operations fU g directly on i logical circuits. This fact means other methods must be encoded data, rather than performing the risky procedure of used to perform universal, fault-tolerant quantum compu- decoding, applying U , and reencoding. In fact, the latter i tation. The most common approach is that of magic procedure is forbidden if we insist on each C being a i states [5,6], encoded ancilla qubits that, combined with fault-tolerant logical circuit, for which the failure of any available transversal circuits (usually implementing one component in C never leads to an uncorrectable error i Clifford operations), serve to complete a universal set of on the encoded data. If, in addition, the set fU g is i logical circuits (usually by implementing a T or Toffoli universal for the computational model in question (e.g., gate). Ideally, these magic states would be efficiently classical or quantum computation), the set of logical constructible themselves, but current so-called distillation circuits fCig is said to be universal, and the error-correcting code in question could, in principle, be used for all procedures actually incur large overheads in terms of time computational purposes without ever needing to decode, and qubits [7,8]. an essential ability for quantum computing especially, It is, therefore, fortunate that other approaches to bypass the universal-transversal no-go theorem exist. Some where decoherence remains the bane of all practical — implementations of quantum algorithms. involve code switching the transversal circuits on two In the quantum computational model, the one paradigm codes together might complete a universal set, encouraging for designing fault-tolerant logical gates that is most development of a method to exchange data between the two preferred is transversality. A logical circuit is transversal code spaces. Simple procedures have been devised for if all physical qubits have interacted with at most one conversions between specific codes, such as between the physical qubit from each code block, and if such a 5-qubit and 7-qubit [9] and between quantum Reed-Muller design preserves the code space, it is automatically fault codes [10]. Another workaround for the no-go theorem tolerant. Indeed, many quantum codes, such as Steane’s uses triorthogonal subsystem codes [11], the smallest of 7-qubit code [1], are highly regarded exactly because which is 15-qubit, to implement a universal, transversal set of gates without code switching, but with additional error correction (EC) on the gauge qubits. Related gauge-fixing techniques are employed in topological color Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- codes [12], where getting locally implementable non- bution of this work must maintain attribution to the author(s) and Clifford gates requires jumping up a dimension to 3D the published article’s title, journal citation, and DOI. [13]. Lastly, concatenated coding combines two codes with 2160-3308=16=6(3)=031039(23) 031039-1 Published by the American Physical Society YODER, TAKAGI, and CHUANG PHYS. REV. X 6, 031039 (2016) complementary transversal gate sets into a single large code latter, logical initialization, measurement, and Clifford with universal, yet not transversal, gates [14]. gates are performed on surface codes by manipulating Here, we complement the myriad previous approaches to the geometry of the surface while correcting errors as they universal fault tolerance by developing nontransversal, yet arise. Both techniques are similar to pieceable fault still fault-tolerant, circuits implementing logical gates on tolerance as the code undergoes transformations to several stabilizer codes. Our approach is based on an under- intermediate codes, each with distance large enough to appreciated trick first used by Knill, Laflamme, and correct any errors that may have arisen. However, both also Zurek [15] to implement a fault-tolerant controlled-S gate do not generate logical universality on their own, some- on the 7-qubit code. The trick revolves around breaking a thing we show pieceable fault tolerance can indeed provide. nontransversal circuit into fault-tolerant pieces, with error Achieving this universality also requires fundamentally correction performed in between to correct errors before new tools. For instance, we develop original circuit designs, they propagate too badly throughout the nontransversal called round-robin constructions, to perform our logical circuit. We show that this trick for creating fault-tolerant gates. Additionally, we create an adaptive procedure for logical gates can be significantly generalized to perform error correction on the nonstabilizer intermediate codes we other logical gates on other codes and ultimately developed encounter. The profitable use of nonstabilizer codes is into a procedure we call pieceable fault tolerance. perhaps novel and interesting in and of itself. A fault- As examples of pieceable fault tolerance, we create a tolerance overview, the definition of pieceable fault toler- logical controlled-Z (CZ) gate on the 5-qubit code (Sec. III) ance, and a summary of these new tools are the goals of the and logical controlled-controlled-Z (CCZ) gates on the next section. 5-qubit (Sec. III) and 7-qubit (Sec. IV) codes, completing universal sets of gates on those codes. Also notable is that II. DESIGN METHODOLOGY FOR our circuits use no ancillas other than those required for the FAULT-TOLERANT LOGICAL GATES multiple rounds of error correction—in particular, we use Quantum codes operate by the “fight entanglement with no magic states. Indeed, in Sec. IV, we find that our entanglement” [21] mantra—to protect sensitive data from construction for the 7-qubit code compares favorably a noisy, nosy environment, introduce additional degrees of against magic-state injection with regards to resources freedom, and encode the data in globally entangled states. required. All our pieceable circuits are 1-fault-tolerant. Local errors have no chance of rearranging the long-range That is, any one fault does not cause a logical error, as is entanglement to affect the data, and so our information is consistent with both of these small codes being distance secure. three. Through concatenation, these pieceable circuits However, if we want to legitimately alter the data, this also possess a fault-tolerance threshold in the usual same security becomes a hassle. To perform quantum gates sense [16,17]. on the encoded data, we are forced to create circuits In addition, we provide sufficient conditions for similar manipulating globally entangled states, and, moreover, pieceable circuits to work on larger codes more generally these circuits must not corrupt the data, even if they (Sec. V). We find that nondegeneracy or, more specifically, themselves are faulty. In this section, we overview this not having weight-two stabilizers, is sufficient (albeit not design challenge, culminating in our solution, pieceably necessary) for a code to have a pieceably 1-fault-tolerant fault-tolerant logical gates. gate. Therefore, any nondegenerate stabilizer code with a universal set of fault-tolerant local (i.e., single-qubit) A. Codes, logical operations, and error correction Clifford logical gates can be promoted to fault-tolerant universality using our pieceable methods. Compare to An ⟦n; k⟧ quantum code L using n physical qubits to magic states, where to achieve the same universality for encode k data qubits is essentially a collection of 2k distance-three codes, even with much more overhead, the orthogonal n-qubit code states fj0¯i; j1¯i; …; j2kig. The code entire set of logical Cliffords is required [5]. Moreover, we space CL is the span of the code states. In this sense, show nondegenerate CSS [1,18] codes just need any fault- quantum codes are nothing more than a strange basis for a tolerant local Clifford to achieve the same universality. subspace of a larger Hilbert space.
Load more

Recommended publications
  • Conversion of a General Quantum Stabilizer Code to an Entanglement
    Conversion of a general quantum stabilizer code to an entanglement distillation protocol∗ Ryutaroh Matsumoto Dept. of Communications and Integrated Systems Tokyo Institute of Technology, 152-8552 Japan Email: [email protected] April 4, 2003 Abstract We show how to convert a quantum stabilizer code to a one-way or two- way entanglement distillation protocol. The proposed conversion method is a generalization of those of Shor-Preskill and Nielsen-Chuang. The recurrence protocol and the quantum privacy amplification protocol are equivalent to the protocols converted from [[2, 1]] stabilizer codes. We also give an example of a two-way protocol converted from a stabilizer bet- ter than the recurrence protocol and the quantum privacy amplification protocol. The distillable entanglement by the class of one-way protocols converted from stabilizer codes for a certain class of states is equal to or greater than the achievable rate of stabilizer codes over the channel cor- responding to the distilled state, and they can distill asymptotically more entanglement from a very noisy Werner state than the hashing protocol. 1 Introduction arXiv:quant-ph/0209091v4 4 Apr 2003 In many applications of quantum mechanics to communication, the sender and the receiver have to share a maximally entangled quantum state of two particles. When there is a noiseless quantum communication channel, the sender can send one of two particles in a maximally entangled state to the receiver and sharing of it is easily accomplished. However, the quantum communication channel is usually noisy, that is, the quantum state of the received particle changes probabilistically from the original state of a particle.
    [Show full text]
  • Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes
    Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes Daniel Litinski and Felix von Oppen Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany We present a planar surface-code-based bilizers requires more potentially faulty controlled-not scheme for fault-tolerant quantum computation (CNOT) gates. which eliminates the time overhead of single- The main drawback of surface codes in comparison qubit Clifford gates, and implements long-range to color codes is the absence of transversal single-qubit multi-target CNOT gates with a time overhead Clifford gates, i.e., the gates that are products of the that scales only logarithmically with the control- Hadamard gate H and the phase gate S. While the target separation. This is done by replacing transversal Clifford gates of color codes provide them hardware operations for single-qubit Clifford with fast logical H and S gates, defect-based propos- gates with a classical tracking protocol. Inter- als for surface codes [14] implement the H gate via a qubit communication is added via a modified multi-step measurement protocol, and the S gate via a lattice surgery protocol that employs twist de- distilled ancilla qubit. In order to lower the overhead fects of the surface code. The long-range multi- of single-qubit Clifford gates, surface code qubits can target CNOT gates facilitate magic state distil- be encoded using twist defects [15], which are essen- lation, which renders our scheme fault-tolerant tially Majoranas that can be braided via code defor- and universal. mation [16].
    [Show full text]
  • Quantum Error-Correcting Codes by Concatenation QEC11
    Quantum Error-Correcting Codes by Concatenation QEC11 Second International Conference on Quantum Error Correction University of Southern California, Los Angeles, USA December 5–9, 2011 Quantum Error-Correcting Codes by Concatenation Markus Grassl joint work with Bei Zeng Centre for Quantum Technologies National University of Singapore Singapore Markus Grassl – 1– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Why Bei isn’t here Jonathan, November 24, 2011 Markus Grassl – 2– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Overview Shor’s nine-qubit code revisited • The code [[25, 1, 9]] • Concatenated graph codes • Generalized concatenated quantum codes • Codes for the Amplitude Damping (AD) channel • Conclusions • Markus Grassl – 3– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 Shor’s Nine-Qubit Code Revisited Bit-flip code: 0 000 , 1 111 . | → | | → | Phase-flip code: 0 + ++ , 1 . | → | | → | − −− Effect of single-qubit errors on the bit-flip code: X-errors change the basis states, but can be corrected • Z-errors at any of the three positions: • Z 000 = 000 | | “encoded” Z-operator Z 111 = 111 | −| = Bit-flip code & error correction convert the channel into a phase-error ⇒ channel = Concatenation of bit-flip code and phase-flip code yields [[9, 1, 3]] ⇒ Markus Grassl – 4– 07.12.2011 Quantum Error-Correcting Codes by Concatenation QEC11 The Code [[25, 1, 9]] The best single-error correcting code is = [[5, 1, 3]] • C0 Re-encoding each of the 5 qubits with yields = [[52, 1, 32]]
    [Show full text]
  • Lecture 6: Quantum Error Correction and Quantum Capacity
    Lecture 6: Quantum error correction and quantum capacity Mark M. Wilde∗ The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally \quantum" theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum com- munication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem provides the intuition behind quantum error correction. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information|otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H(B) − H(E)|a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain.
    [Show full text]
  • Arxiv:2003.09412V2 [Quant-Ph] 7 Jul 2021 Hadamard-Free Circuits Expose the Structure of the Clifford Group
    Hadamard-free circuits expose the structure of the Clifford group Sergey Bravyi and Dmitri Maslov IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA July 8, 2021 Abstract The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form F1HSF2, where H is a layer of Hadamard gates, S is a permutation of qubits, and Fi are parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed n-qubit Clifford operator in runtime O(n2). The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard- free part and one allowing a circuit depth 9n implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place. 1 Introduction Clifford circuits can be defined as quantum computations by the circuits with Phase (p), Hadamard (h), and cnot gates, applied to a computational basis state, such as |00...0i.
    [Show full text]
  • Quantum Topological Error Correction Codes Are Capable of Improving the Performance of Clifford Gates
    1 Quantum Topological Error Correction Codes Are Capable of Improving the Performance of Clifford Gates Daryus Chandra, Zunaira Babar, Hung Viet Nguyen, Dimitrios Alanis, Panagiotis Botsinis, Soon Xin Ng, and Lajos Hanzo Abstract—The employment of quantum error correction codes under certain conditions we can transform various classes of (QECCs) within quantum computers potentially offers a reli- powerful classical error correction codes into their quantum ability improvement for both quantum computation and com- counterparts, such as Quantum Turbo Codes (QTCs) [8], [9], munications tasks. However, incorporating quantum gates for performing error correction potentially introduces more sources Quantum Low-Density Parity-Check (QLDPC) codes [10], of quantum decoherence into the quantum computers. In this [11], and Quantum Polar Codes (QPCs) [12], [13]. However, scenario, the primary challenge is to find the sufficient condition several challenges remain, hindering the immediate employ- required by each of the quantum gates for beneficially employing ment of these powerful QSCs in quantum computers. Firstly, QECCs in order to yield reliability improvements given that the the reliability of the state-of-the-art quantum gates is still quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by significantly lower compared to classical gates. For exam- firstly presenting the general framework of protecting quantum ple, the reliability of a two-qubit quantum gate is between gates by the amalgamation of the transversal configuration of 90:00% 99:90% across various technology platforms, such quantum gates and quantum stabilizer codes (QSCs), which can as spin− electronics, photonics, superconducting, trapped-ion, be viewed as syndrome-based QECCs.
    [Show full text]
  • Performance of Various Low-Level Decoder for Surface Codes in the Presence of Measurement Error
    Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-2021 Performance of Various Low-level Decoder for Surface Codes in the Presence of Measurement Error Claire E. Badger Follow this and additional works at: https://scholar.afit.edu/etd Part of the Computer Sciences Commons Recommended Citation Badger, Claire E., "Performance of Various Low-level Decoder for Surface Codes in the Presence of Measurement Error" (2021). Theses and Dissertations. 4885. https://scholar.afit.edu/etd/4885 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Performance of Various Low-Level Decoders for Surface Codes in the Presence of Measurement Error THESIS Claire E Badger, B.S.C.S., 2d Lieutenant, USAF AFIT-ENG-MS-21-M-008 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio DISTRIBUTION STATEMENT A APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. The views expressed in this document are those of the author and do not reflect the official policy or position of the United States Air Force, the United States Department of Defense or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. AFIT-ENG-MS-21-M-008 Performance of Various Low-Level Decoders for Surface Codes in the Presence of Measurement Error THESIS Presented to the Faculty Department of Electrical and Computer Engineering Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Science Claire E Badger, B.S.C.S., B.S.E.E.
    [Show full text]
  • Universal Quantum Computation with Ideal Clifford Gates and Noisy Ancillas
    PHYSICAL REVIEW A 71, 022316 ͑2005͒ Universal quantum computation with ideal Clifford gates and noisy ancillas Sergey Bravyi* and Alexei Kitaev† Institute for Quantum Information, California Institute of Technology, Pasadena, 91125 California, USA ͑Received 6 May 2004; published 22 February 2005͒ We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state ͉0͘, and qubit measurement in the computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state ␳, which should be regarded as a parameter of the model. Our goal is to determine for which ␳ universal quantum computation ͑UQC͒ can be efficiently simu- lated. To answer this question, we construct purification protocols that consume several copies of ␳ and produce a single output qubit with higher polarization. The protocols allow one to increase the polarization only along certain “magic” directions. If the polarization of ␳ along a magic direction exceeds a threshold value ͑about 65%͒, the purification asymptotically yields a pure state, which we call a magic state. We show that the Clifford group operations combined with magic states preparation are sufficient for UQC. The connection of our results with the Gottesman-Knill theorem is discussed. DOI: 10.1103/PhysRevA.71.022316 PACS number͑s͒: 03.67.Lx, 03.67.Pp I. INTRODUCTION AND SUMMARY additional operations ͑e.g., measurements by Aharonov- Bohm interference ͓13͔ or some gates that are not related to The theory of fault-tolerant quantum computation defines topology at all͒. Of course, these nontopological operations an important number called the error threshold.
    [Show full text]
  • Randomized Benchmarking of Two-Qubit Gates
    Master's Thesis Randomized Benchmarking of Two-Qubit Gates Department of Physics Laboratory for Solid State Physics Quantum Device Lab August 28, 2015 Author: Samuel Haberthur¨ Supervisor: Yves Salathe´ Principal Investor: Prof. Dr. Andreas Wallraff Abstract On the way to high-fidelity quantum gates, accurate estimation of the gate errors is essential. Randomized benchmarking (RB) provides a tool to classify and characterize the errors of multi- qubit gates. Here we implement RB to investigate the fidelity of single-qubit and two-qubit gates in the Clifford group realized on a superconducting transmon qubit system. This thesis provides an overview of the current randomized benchmarking methods and gives detailed discussions of how to estimate the gate error. For single-qubit gates an average gate fidelity of 99:6(1) % was obtained. Furthermore, it shows how two-qubit gates are realized and how they can be calibrated to achieve high fidelities. Here, the focus lies on the controlled phase gate and iswap gate, which are both implemented using fast magnetic flux pulses. For the former, fidelities above 92 % were estimated. Also a decreasing of average gate fidelity over time was observed. Finally, a method for achieving scalable iSwap gates is proposed. Contents 1. Motivation 6 2. Superconducting Quantum Circuits8 2.1. Electromagnetic Oscillators..............................8 2.2. The Transmon Qubit.................................. 11 2.3. Circuit QED...................................... 13 2.4. Readout and Single Quantum Gates......................... 15 2.5. Coherence Time.................................... 17 2.6. Experimental Setup.................................. 18 3. Single-Qubit Randomized Benchmarking 21 3.1. Randomized Noise Estimation............................. 22 3.2. Pauli Method...................................... 23 3.3.
    [Show full text]
  • UC Riverside UC Riverside Electronic Theses and Dissertations
    UC Riverside UC Riverside Electronic Theses and Dissertations Title Codeword Stabilized Quantum Codes and Their Error Correction Permalink https://escholarship.org/uc/item/67h4j8p8 Author Li, Yunfan Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA RIVERSIDE Codeword Stabilized Quantum Codes and Their Error Correction A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Electrical Engineering by Yunfan Li August 2010 Dissertation Committee: Dr. Ilya Dumer, Chairperson Dr. Leonid Pryadko Dr. Alexander N. Korotkov Copyright by Yunfan Li 2010 The Dissertation of Yunfan Li is approved: Committee Chairperson University of California, Riverside Acknowledgments I would like to thank all people who have helped and inspired me during my doctoral study. I especially want to thank my advisor, Professor Ilya Dumer, for his important and valuable guidance during my research and study at UC Riverside. His perpetual energy and enthusiasm in research had motivated me. In addition, he was always willing to help his students with their research. As a result, research life became smooth and rewarding for me. I was delighted to interact with Professor Leonid Pryadko by attending his classes and having him as my co-advisor. I want to thank him for his guidance and joint effort in my research. I would like to thank Professor Alexander N. Korotkov. I got deep understanding about quantum computation and quantum error correcting codes from his lectures. I also want to express my thanks to Bei Zeng for the detailed explanation of the CWS code construction, to Markus Grassl for the important joint effort.
    [Show full text]
  • Arxiv:2006.01273V1 [Quant-Ph] 1 Jun 2020 Performance of the System According to Our Bench- 5.1 Full Stack Benchmarking
    Application-Motivated, Holistic Benchmarking of a Full Quantum Computing Stack Daniel Mills∗ 1,2, Seyon Sivarajah2, Travis L. Scholten3, and Ross Duncan2,4 1University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, UK 2Cambridge Quantum Computing Ltd, 9a Bridge Street, Cambridge, CB2 1UB, UK 3IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA 4University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH, UK June 3, 2020 Abstract Contents Quantum computing systems need to be bench- 1 Introduction2 marked in terms of practical tasks they would be expected to do. Here, we propose 3 “application- 2 Circuit Classes4 motivated” circuit classes for benchmarking: deep 2.1 Shallow Circuits: IQP . .4 (relevant for state preparation in the variational 2.2 Square Circuits: Random Circuit quantum eigensolver algorithm), shallow (inspired Sampling . .5 by IQP-type circuits that might be useful for near- 2.3 Deep Circuits: Pauli Gadgets . .7 term quantum machine learning), and square (in- spired by the quantum volume benchmark). We 3 Figures of Merit7 quantify the performance of a quantum computing 3.1 Heavy Output Generation Bench- system in running circuits from these classes using marking . .8 several figures of merit, all of which require expo- 3.2 Cross-Entropy Difference . 10 nential classical computing resources and a polyno- 3.3 `1-Norm Distance . 11 mial number of classical samples (bitstrings) from 3.4 Metric Comparison . 12 the system. We study how performance varies with the compilation strategy used and the de- 4 Quantum Computing Stack 12 vice on which the circuit is run. Using systems 4.1 Software Development Kits .
    [Show full text]
  • Supercomputer Simulations of Transmon Quantum Computers Quantum Simulations of Transmon Supercomputer
    IAS Series IAS Supercomputer simulations of transmon quantum computers quantum simulations of transmon Supercomputer 45 Supercomputer simulations of transmon quantum computers Dennis Willsch IAS Series IAS Series Band / Volume 45 Band / Volume 45 ISBN 978-3-95806-505-5 ISBN 978-3-95806-505-5 Dennis Willsch Schriften des Forschungszentrums Jülich IAS Series Band / Volume 45 Forschungszentrum Jülich GmbH Institute for Advanced Simulation (IAS) Jülich Supercomputing Centre (JSC) Supercomputer simulations of transmon quantum computers Dennis Willsch Schriften des Forschungszentrums Jülich IAS Series Band / Volume 45 ISSN 1868-8489 ISBN 978-3-95806-505-5 Bibliografsche Information der Deutschen Nationalbibliothek. Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografe; detaillierte Bibliografsche Daten sind im Internet über http://dnb.d-nb.de abrufbar. Herausgeber Forschungszentrum Jülich GmbH und Vertrieb: Zentralbibliothek, Verlag 52425 Jülich Tel.: +49 2461 61-5368 Fax: +49 2461 61-6103 [email protected] www.fz-juelich.de/zb Umschlaggestaltung: Grafsche Medien, Forschungszentrum Jülich GmbH Titelbild: Quantum Flagship/H.Ritsch Druck: Grafsche Medien, Forschungszentrum Jülich GmbH Copyright: Forschungszentrum Jülich 2020 Schriften des Forschungszentrums Jülich IAS Series, Band / Volume 45 D 82 (Diss. RWTH Aachen University, 2020) ISSN 1868-8489 ISBN 978-3-95806-505-5 Vollständig frei verfügbar über das Publikationsportal des Forschungszentrums Jülich (JuSER) unter www.fz-juelich.de/zb/openaccess. This is an Open Access publication distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We develop a simulator for quantum computers composed of superconducting transmon qubits.
    [Show full text]