356 research outputs found
Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates
No full textStabilizer codes are among the most successful quantum error-correcting codes, yet they have important limitations on their ability to fault tolerantly compute. Here, we introduce a new quantity, the disjointness of the stabilizer code, which, roughly speaking, is the number of mostly nonoverlapping representations of any given nontrivial logical Pauli operator. The notion of disjointness proves useful in limiting transversal gates on any error-detecting stabilizer code to a finite level of the Clifford hierarchy. For code families, we can similarly restrict logical operators implemented by constant-depth circuits. For instance, we show that it is impossible, with a constant-depth but possibly geometrically nonlocal circuit, to implement a logical non-Clifford gate on the standard two-dimensional surface code. Subject Areas: Quantum Physics, Quantum InformationAmerican Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipInternational Business Machines Corporation (Ph.D. Fellowship
Error rates and resource overheads of encoded three-qubit gates
No full textA non-Clifford gate is required for universal quantum computation, and, typically, this is the most error-prone and resource-intensive logical operation on an error-correcting code. Small, single-qubit rotations are popular choices for this non-Clifford gate, but certain three-qubit gates, such as Toffoli or controlled-controlled-Z (ccz), are equivalent options that are also more suited for implementing some quantum algorithms, for instance, those with coherent classical subroutines. Here, we calculate error rates and resource overheads for implementing logical ccz with pieceable fault tolerance, a nontransversal method for implementing logical gates. We provide a comparison with a nonlocal magic-state scheme on a concatenated code and a local magic-state scheme on the surface code. We find the pieceable fault-tolerance scheme particularly advantaged over magic states on concatenated codes and in certain regimes over magic states on the surface code. Our results suggest that pieceable fault tolerance is a promising candidate for fault tolerance in a near-future quantum computer
Fixed-point adiabatic quantum search
No full textFixed-point quantum search algorithms succeed at finding one of M target items among N total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property—the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.American Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipMIT-Harvard Center for Ultracold Atoms MIT International Science and Technology InitiativeNational Science Foundation (U.S.) (RQCC Project 1111337)Massachusetts Institute of Technology. Undergraduate Research Opportunities Program (Paul E. Gray Endowed Fund
Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes
No full textIt 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.National Science Foundation (U.S.) (RQCC Project 1111337)United States. Army Research OfficeAmerican Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipTakenaka ScholarshipMIT Department of Physics (Frank Fellowship
Theodore Deppe, 38th Annual ODU Literary Festival
No full textTheodore Deppe is the author of Beautiful Wheel, Orpheus on the Red Line, Cape Clear: New and Selected Poems, The Wanderer King, and Children of the Air. He has received a Pushcart Prize, two fellowships from the National Endowment for the Arts, and grants from the Massachusetts Cultural Commission and the Connecticut Commission on the Arts. His work has appeared in Poetry, Harper’s, Kenyon Review, Ploughshares, Poetry Ireland Review and elsewhere. He served as writer-in-residence at Phillips Academy in Massachusetts, the Poet\u27s House in Ireland, Westminster College in Utah, and the James Merrill House in Connecticut. He directs Stonecoast in Ireland and lives on the west coast of Ireland
Robust calibration of a universal single-qubit gate set via robust phase estimation
No full textAn important step in building a quantum computer is calibrating experimentally implemented quantum gates to produce operations that are close to ideal unitaries. The calibration step involves estimating the systematic errors in gates and then using controls to correct the implementation. Quantum process tomography is a standard technique for estimating these errors but is both time consuming (when one wants to learn only a few key parameters) and usually inaccurate without resources such as perfect state preparation and measurement, which might not be available. With the goal of efficiently and accurately estimating specific errors using minimal resources, we develop a parameter estimation technique, which can gauge key systematic parameters (specifically, amplitude and off-resonance errors) in a universal single-qubit gate set with provable robustness and efficiency. In particular, our estimates achieve the optimal efficiency, Heisenberg scaling, and do so without entanglement and entirely within a single-qubit Hilbert space. Our main theorem making this possible is a robust version of the phase estimation procedure of Higgins et al. [B. L. Higgins et al., New J. Phys. 11, 073023 (2009)NJOPFM1367-263010.1088/1367-2630/11/7/073023].United States. Dept. of DefenseUnited States. Army Research Office. Quantum Algorithms ProgramAmerican Society for Engineering Education. National Defense Science and Engineering Graduate Fellowshi
Fixed-Point Quantum Search with an Optimal Number of Queries
No full textGrover’s quantum search and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction but, as a consequence, lose the very quadratic advantage that makes Grover’s algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ.National Science Foundation (U.S.) (RQCC Project 1111337)United States. Army Research Office (Quantum Algorithms Program)National Science Foundation (U.S.). Integrative Graduate Education and Research Traineeship (Interdisciplinary Quantum Information Science and Engineering Integrative Graduate Education and Research Traineeship
Quantum inference on Bayesian networks
No full textPerforming exact inference on Bayesian networks is known to be #P-hard. Typically approximate inference techniques are used instead to sample from the distribution on query variables given the values e of evidence variables. Classically, a single unbiased sample is obtained from a Bayesian network on n variables with at most m parents per node in time O(nmP(e)[superscript −1]), depending critically on P(e), the probability that the evidence might occur in the first place. By implementing a quantum version of rejection sampling, we obtain a square-root speedup, taking O(n2[superscript m]P(e)[superscript −1/2]) time per sample. We exploit the Bayesian network's graph structure to efficiently construct a quantum state, a q-sample, representing the intended classical distribution, and also to efficiently apply amplitude amplification, the source of our speedup. Thus, our speedup is notable as it is unrelativized—we count primitive operations and require no blackbox oracle queries.United States. Army Research Office (Project W911NF1210486)National Science Foundation (U.S.). Integrative Graduate Education and Research TraineeshipNational Science Foundation (U.S.). Center for Ultracold Atom
Optimal arbitrarily accurate composite pulse sequences
No full textImplementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors ε, caused by incorrect duration or strength of a pulse, are an especially common problem. But a sequence of imperfect pulses can provide a better implementation of a desired operation, as compared to a single primitive pulse. We find optimal pulse sequences consisting of L primitive π or 2π rotations that suppress such errors to arbitrary order O(ε[superscript n]) on arbitrary initial states. Optimality is demonstrated by proving an L = O(n) lower bound and saturating it with L = 2n solutions. Closed-form solutions for arbitrary rotation angles are given for n = 1,2,3,4. Perturbative solutions for any n are proven for small angles, while arbitrary angle solutions are obtained by analytic continuation up to n = 12. The derivation proceeds by a novel algebraic and nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.National Science Foundation (U.S.). Center for Ultracold Atoms (1125846)National Science Foundation (U.S.) (RQCC 1111337)National Science Foundation (U.S.) (iQuISE IGERT)United States. Intelligence Advanced Research Projects Activity (QCS ORAQL project
Methodology of Resonant Equiangular Composite Quantum Gates
No full textThe creation of composite quantum gates that implement quantum response functions [^ over U](θ) dependent on some parameter of interest θ is often more of an art than a science. Through inspired design, a sequence of L primitive gates also depending on θ can engineer a highly nontrivial [^ over U](θ) that enables myriad precision metrology, spectroscopy, and control techniques. However, discovering new, useful examples of [^ over U](θ) requires great intuition to perceive the possibilities, and often brute force to find optimal implementations. We present a systematic and efficient methodology for composite gate design of arbitrary length, where phase-controlled primitive gates all rotating by θ act on a single spin. We fully characterize the realizable family of [^ over U](θ), provide an efficient algorithm that decomposes a choice of [^ over U](θ) into its shortest sequence of gates, and show how to efficiently choose an achievable [^ over U](θ) that, for fixed L, is an optimal approximation to objective functions on its quadratures. A strong connection is forged with classical discrete-time signal processing, allowing us to swiftly construct, as examples, compensated gates with optimal bandwidth that implement arbitrary single-spin rotations with subwavelength spatial selectivity.National Science Foundation (U.S.) (RQCC Project 1111337)American Society for Engineering Education. National Defense Science and Engineering Graduate FellowshipUnited States. National Reconnaissance Offic
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