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Building a quantum computer that surpasses the computational power of its classical counterpart is a great engineering challenge. Quantum software optimizations can provide an accelerated pathway to the first generation of quantum computing applications that might save years of engineering effort. Current quantum software stacks follow a layered approach similar to the stack of classical computers, which was designed to manage the complexity. In this review, we point out that greater efficiency of quantum computing systems can be achieved by breaking the abstractions between these layers. We review several works along this line, including two hardware-aware compilation optimizations that break the quantum Instruction Set Architecture (ISA) abstraction and two error-correction/information-processing schemes that break the qubit abstraction. Last, we discuss several possible future directions., Invited paper by Proceedings of IEEE special issue

Leakage is a particularly damaging error that occurs when a qubit leaves the defined computational subspace. Leakage errors limit the effectiveness of quantum error correcting codes by spreading additional errors to other qubits and corrupting syndrome measurements. The effects of leakage errors on the surface code has been studied in various contexts. However, the effects of a leaked data qubit versus a leaked ancilla qubit can be quite different. Here, we study the effects of data leakage and ancilla leakage separately. We show that data leakage is much less damaging. We show that the surface code maintains its distance in the presence of leakage by either confining leakage to data qubits or eliminating aniclla qubit leakage at the critical fault location. We also introduce new techniques for handling leakage by using gates with one-sided leakage and by mixing two types of leakage reducing circuits: one to handle data leakage and one to handle ancilla leakage., Comment: comments welcome

We introduce a single-number metric, quantum volume, that can be measured using a concrete protocol on near-term quantum computers of modest size ($n\lesssim 50$), and measure it on several state-of-the-art transmon devices, finding values as high as 16. The quantum volume is linked to system error rates, and is empirically reduced by uncontrolled interactions within the system. It quantifies the largest random circuit of equal width and depth that the computer successfully implements. Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting toolchains are expected to have higher quantum volumes. The quantum volume is a pragmatic way to measure and compare progress toward improved system-wide gate error rates for near-term quantum computation and error-correction experiments., Comment: 12 pages, 8 figures, closer to published version

In this work we introduce two code families, which we call the heavy hexagon code and heavy square code. Both code families are implemented by assigning physical data and ancilla qubits to both vertices and edges of low degree graphs. Such a layout is particularly suitable for superconducting qubit architectures to minimize frequency collisions and crosstalk. In some cases, frequency collisions can be reduced by several orders of magnitude. The heavy hexagon code is a hybrid surface/Bacon-Shor code mapped onto a (heavy) hexagonal lattice whereas the heavy square code is the surface code mapped onto a (heavy) square lattice. In both cases, the lattice includes all the ancilla qubits required for fault-tolerant error-correction. Naively, the limited qubit connectivity might be thought to limit the error-correcting capability of the code to less than its full distance. Therefore, essential to our construction is the use of flag qubits. We modify minimum weight perfect matching decoding to efficiently and scalably incorporate information from measurements of the flag qubits and correct up to the full code distance while respecting the limited connectivity. Simulations show that high threshold values for both codes can be obtained using our decoding protocol. Further, our decoding scheme can be adapted to other topological code families., 20 pages, 21 figures, Comments welcome! V2 conforms to journal specifications

The concept of quantum computing has inspired a whole new generation of scientists, including physicists, engineers, and computer scientists, to fundamentally change the landscape of information technology. With experimental demonstrations stretching back more than two decades, the quantum computing community has achieved a major milestone over the past few years: the ability to build systems that are stretching the limits of what can be classically simulated, and which enable cloud-based research for a wide range of scientists, thus increasing the pool of talent exploring early quantum systems. While such noisy near-term quantum computing systems fall far short of the requirements for fault-tolerant systems, they provide unique test beds for exploring the opportunities for quantum applications. Here, we highlight an IBM-specific perspective of the facets associated with these systems, including quantum software, cloud access, benchmarking quantum systems, error correction and mitigation in such systems, understanding the complexity of quantum circuits, and how early quantum applications can run on near-term quantum computers.

Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit superconducting transmon device and demonstrate fault-tolerant state preparation. We characterize the resulting code words through quantum process tomography and study the free evolution of the logical observables. Our results are consistent with fault-tolerant state preparation in a protected qubit subspace.

Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed. With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupled-cluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits., Comment: Contribution to the special issue of Quantum Science & Technology on "What would you do with 1000 qubits"

We present a unifying approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. We use this framework to generate new codes with superior parameters to any previously known. In particular, we find ((10,18,3)) and ((10,20,3)) codes. We also show how to construct encoding circuits for all codes within our framework., 5 pages, 1 eps figure, ((11,48,3)) code removed, encoding circuits added, typos corrected in codewords and elsewhere

We summarize the main characteristics of the quantum logic array (QLA) architecture with a careful look at the key issues not described in the original conference publications: primarily, the teleportation-based logical interconnect. The design goal of the the quantum logic array architecture is to illustrate a model for a large-scale quantum architecture that solves the primary challenges of system-level reliability and data distribution over large distances. The QLA's logical interconnect design, which employs the quantum repeater protocol, is in principle capable of supporting the communication requirements for applications as large as the factoring of a 2048-bit number using Shor's quantum factoring algorithm. Our physical-level assumptions and architectural component validations are based on the trapped ion technology for implementing quantum computing.

Information theory establishes the fundamental limits on data transmission, storage, and processing. Quantum information theory unites information theoretic ideas with an accurate quantum-mechanical description of reality to give a more accurate and complete theory with new and more powerful possibilities for information processing. The goal of both classical and quantum information theory is to quantify the optimal rates of interconversion of different resources. These rates are usually characterized in terms of entropies. However, nonadditivity of many entropic formulas often makes finding answers to information theoretic questions intractable. In a few auspicious cases, such as the classical capacity of a classical channel, the capacity region of a multiple access channel and the entanglement assisted capacity of a quantum channel, additivity allows a full characterization of optimal rates. Here we present a new mathematical property of entropic formulas, uniform additivity, that is both easily evaluated and rich enough to capture all known quantum additive formulas. We give a complete characterization of uniformly additive functions using the linear programming approach to entropy inequalities. In addition to all known quantum formulas, we find a new and intriguing additive quantity: the completely coherent information. We also uncover a remarkable coincidence---the classical and quantum uniformly additive functions are identical; the tractable answers in classical and quantum information theory are formally equivalent. Our techniques pave the way for a deeper understanding of the tractability of information theory, from classical multi-user problems like broadcast channels to the evaluation of quantum channel capacities., Comment: 13 pages with 4 figures + 25 page appendix

The resonator-induced phase (RIP) gate is a multi-qubit entangling gate that allows a high degree of flexibility in qubit frequencies, making it attractive for quantum operations in large-scale architectures. We experimentally realize the RIP gate with four superconducting qubits in a three-dimensional (3D) circuit-quantum electrodynamics architecture, demonstrating high-fidelity controlled-Z (CZ) gates between all possible pairs of qubits from two different 4-qubit devices in pair subspaces. These qubits are arranged within a wide range of frequency detunings, up to as large as 1.8 GHz. We further show a dynamical multi-qubit refocusing scheme in order to isolate out 2-qubit interactions, and combine them to generate a four-qubit Greenberger-Horne-Zeilinger state., Comment: 5 pages, 4 figures in main text with 9 pages 4 figures in the supplemental material

Noise is often regarded as anathema to quantum computation, but in some settings it can be an unlikely ally. We consider the problem of learning the class of $n$-bit parity functions by making queries to a quantum example oracle. In the absence of noise, quantum and classical parity learning are easy and almost equally powerful, both information-theoretically and computationally. We show that in the presence of noise this story changes dramatically. Indeed, the classical learning problem is believed to be intractable, while the quantum version remains efficient. Depolarizing the qubits at the oracle's output at any constant nonzero rate does not increase the computational (or query) complexity of quantum learning more than logarithmically. However, the problem of learning from corresponding classical examples is the learning parity with noise problem, for which the best known algorithms have superpolynomial complexity. This creates the possibility of observing a quantum advantage with a few hundred noisy qubits. The presence of noise is essential for creating this quantum-classical separation.

An arbitrarily reliable quantum computer can be efficiently constructed from noisy components using a recursive simulation procedure, provided that those components fail with probability less than the fault-tolerance threshold. Recent estimates of the threshold are near some experimentally achieved gate fidelities. However, the landscape of threshold estimates includes pseudothresholds, threshold estimates based on a subset of components and a low level of the recursion. In this paper, we observe that pseudothresholds are a generic phenomenon in fault-tolerant computation. We define pseudothresholds and present classical and quantum fault-tolerant circuits exhibiting pseudothresholds that differ by a factor of $4$ from fault-tolerance thresholds for typical relationships between component failure rates. We develop tools for visualizing how reliability is influenced by recursive simulation in order to determine the asymptotic threshold. Finally, we conjecture that refinements of these methods may establish upper bounds on the fault-tolerance threshold for particular codes and noise models.

We discuss how the presence of gauge sub-systems in the Bacon-Shor code [D. Bacon, Phys. Rev. A 73, 012340 (2006)] leads to remarkably simple and efficient methods for fault-tolerant error correction (FTEC). Most notably, FTEC does not require entangled ancillary states and it can be implemented with nearest-neighbor two-qubit measurements. By using these methods, we prove a lower bound on the quantum accuracy threshold, 1.94 \times 10^{-4} for adversarial stochastic noise, that improves previous lower bounds by nearly an order of magnitude., 4 pages, 2 figures. v3: small revisions, appendix moved to chapter 5 in quant-ph/0703230

Irrespective of the underlying technology used to implement a large-scale quantum architecture system, one of the central challenges of accurately modeling the architecture is the ability to map and schedule a quantum application onto a physical grid while taking into account the cost of communication, the classical resources, and the maximum exploitable parallelism. In this paper we introduce and evaluate a physical operations scheduler for arbitrary quantum circuits. Our scheduler accepts a description of a circuit together with a description of a specific physical layout and outputs a sequence of operations that expose the required communication and available parallelism in the circuit. The output of the scheduler is a quantum assembly language file that can directly be simulated on a set of available tools.

The assumption of maximum parallelism support for the successful realization of scalable quantum computers has led to homogeneous, ``sea-of-qubits'' architectures. The resulting architectures overcome the primary challenges of reliability and scalability at the cost of physically unacceptable system area. We find that by exploiting the natural serialization at both the application and the physical microarchitecture level of a quantum computer, we can reduce the area requirement while improving performance. In particular we present a scalable quantum architecture design that employs specialization of the system into memory and computational regions, each individually optimized to match hardware support to the available parallelism. Through careful application and system analysis, we find that our new architecture can yield up to a factor of thirteen savings in area due to specialization. In addition, by providing a memory hierarchy design for quantum computers, we can increase time performance by a factor of eight. This result brings us closer to the realization of a quantum processor that can solve meaningful problems., 12 pages, 8 figures, To appear in the International Symposium on Computer Architecture (ISCA-33), Boston, MA, 2006

Recent experimental advances have demonstrated technologies capable of supporting scalable quantum computation. A critical next step is how to put those technologies together into a scalable, fault-tolerant system that is also feasible. We propose a Quantum Logic Array (QLA) microarchitecture that forms the foundation of such a system. The QLA focuses on the communication resources necessary to efficiently support fault-tolerant computations. We leverage the extensive groundwork in quantum error correction theory and provide analysis that shows that our system is both asymptotically and empirically fault tolerant. Specifically, we use the QLA to implement a hierarchical, array-based design and a logarithmic expense quantum-teleportation communication protocol. Our goal is to overcome the primary scalability challenges of reliability, communication, and quantum resource distribution that plague current proposals for large-scale quantum computing., 13 pages, 9 figures, 2 tables. To appear in the 2005 International Symposium on Microarchitecture (MICRO-38)

The codeword stabilized ("CWS") quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:0708.1021). This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we prove three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find there does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank (arXiv:cs/0610159)., Comment: 11 pages, 1 figure