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.