The disjointness of stabilizer codes and limitations on fault-tolerant logical gates

Stabilizer codes are a simple and successful class of quantum error-correcting codes. Yet this success comes in spite of some harsh limitations on the ability of these codes to fault-tolerantly compute. Here we introduce a new metric for these codes, the disjointness, which, roughly speaking, is the number of mostly non-overlapping representatives of any given non-trivial logical Pauli operator. We use the disjointness to prove that transversal gates on error-detecting stabilizer codes are necessarily in a finite level of the Clifford hierarchy. We also apply our techniques to topological code families to find similar bounds on the level of the hierarchy attainable by constant depth circuits, regardless of their geometric locality. For instance, we can show that symmetric 2D surface codes cannot have non-local constant depth circuits for non-Clifford gates.Comment: 8+3 pages, 2 figures. Comments welcom

Universal fault-tolerant gates on concatenated stabilizer codes

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 non-transversal, 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 non-transversality 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.Comment: 18 pages + 5 pages appendix, 12 figure

Quantum imaging by coherent enhancement

Conventional wisdom dictates that to image the position of fluorescent atoms or molecules, one should stimulate as much emission and collect as many photons as possible. That is, in this classical case, it has always been assumed that the coherence time of the system should be made short, and that the statistical scaling $\sim1/\sqrt{t}$ defines the resolution limit for imaging time $t$. However, here we show in contrast that given the same resources, a long coherence time permits a higher resolution image. In this quantum regime, we give a procedure for determining the position of a single two-level system, and demonstrate that the standard errors of our position estimates scale at the Heisenberg limit as $\sim 1/t$, a quadratic, and notably optimal, improvement over the classical case.Comment: 4 pages, 4 figue

Quantum Inference on Bayesian Networks

Performing 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 $\mathcal{O}(nmP(e)^{-1})$, depending critically on $P(e)$, the probability the evidence might occur in the first place. By implementing a quantum version of rejection sampling, we obtain a square-root speedup, taking $\mathcal{O}(n2^mP(e)^{-\frac12})$ 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.Comment: 8 pages, 3 figures. Submitted to PR

Fixed-point quantum search with an optimal number of queries

Grover's quantum search and its generalization, quantum amplitude amplification, provide quadratic advantage over classical algorithms for a diverse set of tasks, but are tricky to use without knowing beforehand what fraction $\lambda$ 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 $\lambda$.Comment: 4 pages plus references, 2 figure