A State Distillation Protocol to Implement Arbitrary Single-qubit Rotations

An important task required to build a scalable, fault-tolerant quantum computer is to efficiently represent an arbitrary single-qubit rotation by fault-tolerant quantum operations. Traditionally, the method for decomposing a single-qubit unitary into a discrete set of gates is Solovay-Kitaev decomposition, which in practice produces a sequence of depth O(\log^c(1/\epsilon)), where c~3.97 is the state-of-the-art. The proven lower bound is c=1, however an efficient algorithm that saturates this bound is unknown. In this paper, we present an alternative to Solovay-Kitaev decomposition employing state distillation techniques which reduces c to between 1.12 and 2.27, depending on the setting. For a given single-qubit rotation, our protocol significantly lowers the length of the approximating sequence and the number of required resource states (ancillary qubits). In addition, our protocol is robust to noise in the resource states.Comment: 10 pages, 18 figures, 5 table

Distillation protocols for Fourier states in quantum computing

Fourier states are multi-qubit registers that facilitate phase rotations in fault-tolerant quantum computing. We propose distillation protocols for constructing the fundamental, $n$-qubit Fourier state with error $O(2^{-n})$ at a cost of $O(n \log n)$ Toffoli gates and Clifford gates, or any arbitrary Fourier state using $O(n^2)$ gates. We analyze these protocols with methods from digital signal processing. These results suggest that phase kickback, which uses Fourier states, could be the current lowest-overhead method for generating arbitrary phase rotations.Comment: 18 pages, 4 figure

Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures

Quantum computers have recently made great strides and are on a long-term path towards useful fault-tolerant computation. A dominant overhead in fault-tolerant quantum computation is the production of high-fidelity encoded qubits, called magic states, which enable reliable error-corrected computation. We present the first detailed designs of hardware functional units that implement space-time optimized magic-state factories for surface code error-corrected machines. Interactions among distant qubits require surface code braids (physical pathways on chip) which must be routed. Magic-state 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 space-time volume compared to the best-known previous designs for magic-state factories.Comment: 13 pages, 10 figure

Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count

We present an infinite family of protocols to distill magic states for $T$-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the $T$-gate depth of the circuit scale linearly in the logarithm of the target error $\delta$ (up to $\log \log 1/\delta$). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.Comment: 39 pages, (v2) renamed "odd" and "even" weakly self-dual CSS codes of (v1) to "normal" and "hyperbolic" codes, respectively. (v3) published in Quantu

Resource Optimized Quantum Architectures for Surface Code Implementations of Magic-State Distillation

Quantum computers capable of solving classically intractable problems are under construction, and intermediate-scale devices are approaching completion. Current efforts to design large-scale devices require allocating immense resources to error correction, with the majority dedicated to the production of high-fidelity ancillary states known as magic-states. Leading techniques focus on dedicating a large, contiguous region of the processor as a single "magic-state distillation factory" responsible for meeting the magic-state demands of applications. In this work we design and analyze a set of optimized factory architectural layouts that divide a single factory into spatially distributed factories located throughout the processor. We find that distributed factory architectures minimize the space-time volume overhead imposed by distillation. Additionally, we find that the number of distributed components in each optimal configuration is sensitive to application characteristics and underlying physical device error rates. More specifically, we find that the rate at which T-gates are demanded by an application has a significant impact on the optimal distillation architecture. We develop an optimization procedure that discovers the optimal number of factory distillation rounds and number of output magic states per factory, as well as an overall system architecture that interacts with the factories. This yields between a 10x and 20x resource reduction compared to commonly accepted single factory designs. Performance is analyzed across representative application classes such as quantum simulation and quantum chemistry.Comment: 16 pages, 14 figure