Reliable qubits are difficult to engineer, but standard faulttolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with faulttolerance schemes with multiple encoded qubits. Here, we study the 15qubit Hamming code, which protects seven encoded qubits to distance three. We give faulttolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 in total. In particular, individual encoded qubits within the code block can be targeted. Faulttolerant universal computation is possible with four extra qubits, 19 in total. The procedures could enable testing more sophisticated protected circuits in smallscale quantum devices. Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance.
 Award ID(s):
 1753240
 Publication Date:
 NSFPAR ID:
 10146601
 Journal Name:
 Quantum
 Volume:
 3
 Page Range or eLocationID:
 180
 ISSN:
 2521327X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract 
Quantum computers have recently made great strides and are on a longterm path towards useful faulttolerant computation. A dominant overhead in faulttolerant quantum computation is the production of highfidelity encoded qubits, called magic states, which enable reliable errorcorrected computation. We present the first detailed designs of hardware functional units that implement spacetime optimized magicstate factories for surface code errorcorrected machines. Interactions among distant qubits require surface code braids (physical pathways on chip) which must be routed. Magicstate factories are circuits comprised of a complex set of braids that is more difficult to route than quantum circuits considered in previous work [1]. This paper explores the impact of scheduling techniques, such as gate reordering and qubit renaming, and we propose two novel mapping techniques: braid repulsion and dipole moment braid rotation. We combine these techniques with graph partitioning and community detection algorithms, and further introduce a stitching algorithm for mapping subgraphs onto a physical machine. Our results show a factor of 5.64 reduction in spacetime volume compared to the bestknown previous designs for magicstate factories.

Adiabatic computing with two degrees of freedom of 2local Hamiltonians has been theoretically shown to be equivalent to the gate model of universal quantum computing. But today’s quantum annealers, namely DWave’s 2000Q platform, only provide a 2local Ising Hamiltonian abstraction with a single degree of freedom. This raises the question what subset of gate programs can be expressed as quadratic unconstrained binary problems (QUBOs) on the DWave. The problem is of interest because gatebased quantum platforms are currently limited to 20 qubits while DWave provides 2,000 qubits. However, when transforming entire gate circuits into QUBOs, additional qubits will be required. The objective of this work is to determine a subset of quantum gates suitable for transformation into singledegree 2local Ising Hamiltonians under a common qubit base representation such that they comprise a compound circuit suitable for pure quantum computation, i.e., without having to switch between classical and quantum computing for different bases. To this end, this work contributes, for the first time, a fully automated method to translate quantum gate circuits comprised of a subset of common gates expressed as an IBM Qiskit program to singledegree 2local Ising Hamiltonians, which are subsequently embedded in the DWave 2000Q chimera graph. Thesemore »

Abstract We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimalweight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and faulttolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is heldmore »

Abstract We prove that
depth local random quantum circuits with two qudit nearestneighbor gates on a$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ $\phantom{\rule{0ex}{0ex}}\text{poly}\phantom{\rule{0ex}{0ex}}\left(t\right)\xb7{n}^{1/D}$D dimensional lattice withn qudits are approximatet designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$${{\,\textrm{poly}\,}}(t)\cdot n$$ $\phantom{\rule{0ex}{0ex}}\text{poly}\phantom{\rule{0ex}{0ex}}\left(t\right)\xb7n$ . We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($$D=1$$ $D=1$ ) is infinite and that certain counting problems are$${{\,\mathrm{\textsf{PH}}\,}}$$ $\phantom{\rule{0ex}{0ex}}\mathrm{PH}\phantom{\rule{0ex}{0ex}}$ hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constantdepth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anticoncentration”, meaning roughly that the output has nearmaximal entropy. Unitary 2designs have the desired anticoncentration property. Our result improves the required depth for this level of anticoncentration from linear depthmore »$$\#{\textsf{P}}$$ $\#P$