Leveraging Automorphisms of Quantum Codes for Fault-Tolerant Quantum Computation

Fault-tolerant quantum computation is a technique that is necessary to build a scalable quantum computer from noisy physical building blocks. Key for the implementation of fault-tolerant computations is the ability to perform a universal set of quantum gates that act on the code space of an underlying quantum code. To implement such a universal gate set fault-tolerantly is an expensive task in terms of physical operations, and any possible shortcut to save operations is potentially beneficial and might lead to a reduction in overhead for fault-tolerant computations. We show how the automorphism group of a quantum code can be used to implement some operators on the encoded quantum states in a fault-tolerant way by merely permuting the physical qubits. We derive conditions that a code has to satisfy in order to have a large group of operations that can be implemented transversally when combining transversal CNOT with automorphisms. We give several examples for quantum codes with large groups, including codes with parameters [[8,3,3]], [[15,7,3]], [[22,8,4]], and [[31,11,5]]

High-threshold and low-overhead fault-tolerant quantum memory

Quantum error correction becomes a practical possibility only if the physical error rate is below a threshold value that depends on a particular quantum code, syndrome measurement circuit, and a decoding algorithm. Here we present an end-to-end quantum error correction protocol that implements fault-tolerant memory based on a family of LDPC codes with a high encoding rate that achieves an error threshold of $0.8\%$ for the standard circuit-based noise model. This is on par with the surface code which has remained an uncontested leader in terms of its high error threshold for nearly 20 years. The full syndrome measurement cycle for a length-$n$ code in our family requires $n$ ancillary qubits and a depth-7 circuit composed of nearest-neighbor CNOT gates. The required qubit connectivity is a degree-6 graph that consists of two edge-disjoint planar subgraphs. As a concrete example, we show that 12 logical qubits can be preserved for ten million syndrome cycles using 288 physical qubits in total, assuming the physical error rate of $0.1\%$. We argue that achieving the same level of error suppression on 12 logical qubits with the surface code would require more than 4000 physical qubits. Our findings bring demonstrations of a low-overhead fault-tolerant quantum memory within the reach of near-term quantum processors

Algoritmos cuánticos tolerantes a fallos

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, leída el 17-01-2023The framework of this thesis is fault-tolerant quantum algorithms, which can roughly be divided into the following non-disjoint families: a) Grover’s algorithm and quantum walks, b) Shor’s algorithm and hidden subgroup problems, c) quantum simulation algorithms, d) quantum linear algebra, and e) variational quantum algorithms. All of them are covered, to some extent, in this thesis. Grover’s algorithm and quantum walks are described in Chapter 2. We start by highlighting the central role that rotations play in quantum algorithms, explaining Grover’s, why it is optimal, and how it may be extended. Key subroutines explained in this area are amplitude amplification and quantum walks, which will constitute useful parts of other algorithms. In this chapter, we present our Ref. [62], where we explore the heuristic use of quantum Metropolis and quantum walk algorithms for solving anNP-hard problem. This method has been suggested as an avenue to digitally simulate quantum annealing and preparing ground states of many-body Hamiltonians. In the third chapter, in contrast, we turn to the exponential advantages promisedby the Fourier transform in the context of the hidden subgroup problem. However, since this application is restricted to cryptography, we later explore its use in quantum linear algebra problems. Here we explain the development of the original quantum linear solver algorithm, its improvements, and finally the dequantization techniques that would often restrict the quantum advantage to polynomial...El marco conceptual de esta tesis son los algoritmos cuánticos tolerantes a fallos, que pueden dividirse aproximadamente en las siguientes clases no mutuamente excluyentes :a) algoritmo de Grover y paseos cuánticos, b) algoritmo de Shor y problemas de subgrupos ocultos, c) algoritmos de simulación cuántica, d) álgebra lineal cuántica, ye) algoritmos cuánticos variacionales. Todos ellos se tratan, en cierta medida, en esta tesis. El algoritmo de Grover y los paseos cuánticos se explican en el capítulo 2. Comenzamos destacando el papel central que juegan las rotaciones en los algoritmos cuánticos, explicando el de Grover, por qué es óptimo, y cómo puede ser extendido. Las subrutinas clave explicadas en esta área son la amplificación de la amplitud y los paseos cuánticos, que serán partes importantes de otros algoritmos. En este capítulo presentamos nuestra Ref. [62], donde exploramos el uso heurístico de los algoritmos de Metrópolis y paseos cuánticos para resolver problemas NP-difíciles. De hecho, este método ha sido sugerido como una vía para simular digitalmente el método conocido como ‘quantum annealing’,y la preparación de estados fundamentales de Hamiltonianos ‘many-body’.En el tercer capítulo, en cambio, nos centramos en las ventajas exponenciales que promete la transformada de Fourier en el contexto del problema de los subgrupos ocultos. Sin embargo, dado que esta aplicación está restringida a la criptografía, más adelante exploramos su uso en problemas de álgebra lineal cuántica. Aquí explicamos el desarrollo del algoritmo cuántico original para la resolución de sistemas lineales de ecuaciones, sus mejoras, y finalmente las técnicas de ‘descuantización’ que a menudo restringen la ventaja cuántica a polinómica...Fac. de Ciencias FísicasTRUEunpu

Modeling and managing noise in quantum error correction

Simulating a quantum system to full accuracy is very costly and often impossible as we do not know the exact dynamics of a given system. In particular, the dynamics of measurement noise are not well understood. For this reason, and especially in the context of quantum error correction, where we are studying a larger system with branching outcomes due to syndrome measurement, studies often assume a probabilistic Pauli (or Weyl) noise model on the system with probabilistically misreported outcomes for the measurements. In this thesis, we explore methods to decrease the computational complexity of simulating encoded memory channels by deriving conditions under which effective channels are equivalent up to logical operations. Leveraging this method allows for a significant reduction in computational complexity when simulating quantum error correcting codes. We then propose methods to enforce a model consistent with the typical assumptions of stochastic Pauli (or Weyl) noise with probabilistically misreported measurement outcomes: first via a new protocol we call measurement randomized compiling, which enforces an average noise on measurements wherein measurement outcomes are probabilistically misreported; then by another new protocol we call logical randomized compiling, which enforces the same model on syndrome measurements and a probabilistic Pauli (or Weyl) noise model on all other operations (including idling). Together, these results enable more efficient simulation of quantum error correction systems by enforcing effective noise of a form which is easier to model and by reducing the simulation overhead further via symmetries. The enforced effective noise model is additionally consistent with standard error correction procedures and enables techniques founded upon the standard assumptions to be applied in any setting where our protocols are simultaneously applied