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Register a new userA tree tensor network approach to simulating Shors algorithm
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Authors
Eugene Dumitrescu
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Simulating quantum systems constructively furthers our understanding of qualitative and quantitative features which may be analytically intractable. In this letter, we directly simulate and explore the entanglement structure present in a paradigmatic example of quantum information: Shors wavefunction. The methodology employed is a dynamical tensor network which is initially constructed as a tree tensor network, inspired by the modular exponentiation quantum circuit, and later efficiently mapped to a matrix product state. Utilizing the Schmidt number as a local entanglement metric, our construction explicitly captures the wavefunctions non-local entanglement structure and an entanglement scaling relation is discovered. Specifically, we see that entanglement across a bipartition grows exponentially in the number of qubits before saturating at a critical scale which is proportional to the modular periodicity.
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Shors algorithm is examined critically from the standpoint of its eventual use to obtain the factors of large integers.
We show how the execution time of algorithms on quantum computers depends on the architecture of the quantum computer, the choice of algorithms (including subroutines such as arithmetic), and the ``clock speed of the quantum computer. The primary architectural features of interest are the ability to execute multiple gates concurrently, the number of application-level qubits available, and the interconnection network of qubits. We analyze Shors algorithm for factoring large numbers in this context. Our results show that, if arbitrary interconnection of qubits is possible, a machine with an application-level clock speed of as low as one-third of a (possibly encoded) gate per second could factor a 576-bit number in under one month, potentially outperforming a large network of classical computers. For nearest-neighbor-only architectures, a clock speed of around twenty-seven gates per second is required.
We describe a quantum-assisted machine learning (QAML) method in which multivariate data is encoded into quantum states in a Hilbert space whose dimension is exponentially large in the length of the data vector. Learning in this space occurs through applying a low-depth quantum circuit with a tree tensor network (TTN) topology, which acts as an unsupervised feature extractor to identify the most relevant quantum states in a data-driven fashion. This unsupervised feature extractor then feeds a supervised linear classifier and encodes the output in a small-dimensional quantum register. In contrast to previous work on emph{quantum-inspired} TTN classifiers, in which the embedding map and class decision weights did not map the data to well-defined quantum states, we present an approach that can be implemented on gate-based quantum computing devices. In particular, we identify an embedding map with accuracy similar to exponential machines (Novikov emph{et al.}, arXiv:1605.03795), but which produces valid quantum states from classical data vectors, and utilize manifold-based gradient optimization schemes to produce isometric operations mapping quantum states to a register of qubits defining a class decision. We detail methods for efficiently obtaining one- and two-point correlation functions of the decision boundary vectors of the quantum model, which can be used for model interpretability, as well as methods for obtaining classifications from partial data vectors. Further, we show that the use of isometric tensors can significantly aid in the human interpretability of the correlation functions extracted from the decision weights, and may produce models that are less susceptible to adversarial perturbations. We demonstrate our methodologies in applications utilizing the MNIST handwritten digit dataset and a multivariate timeseries dataset of human activity recognition.
We optimize the area and latency of Shors factoring while simultaneously improving fault tolerance through: (1) balancing the use of ancilla generators, (2) aggressive optimization of error correction, and (3) tuning the core adder circuits. Our custom CAD flow produces detailed layouts of the physical components and utilizes simulation to analyze circuits in terms of area, latency, and success probability. We introduce a metric, called ADCR, which is the probabilistic equivalent of the classic Area-Delay product. Our error correction optimization can reduce ADCR by an order of magnitude or more. Contrary to conventional wisdom, we show that the area of an optimized quantum circuit is not dominated exclusively by error correction. Further, our adder evaluation shows that quantum carry-lookahead adders (QCLA) beat ripple-carry adders in ADCR, despite being larger and more complex. We conclude with what we believe is one of most accurate estimates of the area and latency required for 1024-bit Shors factorization: 7659 mm$^{2}$ for the smallest circuit and $6 * 10^8$ seconds for the fastest circuit.
The quantum multicomputer consists of a large number of small nodes and a qubus interconnect for creating entangled state between the nodes. The primary metric chosen is the performance of such a system on Shors algorithm for factoring large numbers: specifically, the quantum modular exponentiation step that is the computational bottleneck. This dissertation introduces a number of optimizations for the modular exponentiation. My algorithms reduce the latency, or circuit depth, to complete the modular exponentiation of an n-bit number from O(n^3) to O(n log^2 n) or O(n^2 log n), depending on architecture. Calculations show that these algorithms are one million times and thirteen thousand times faster, when factoring a 6,000-bit number, depending on architecture. Extending to the quantum multicomputer, five different qubus interconnect topologies are considered, and two forms of carry-ripple adder are found to be the fastest for a wide range of performance parameters. The links in the quantum multicomputer are serial; parallel links would provide only very modest improvements in system reliability and performance. Two levels of the Steane [[23,1,7]] error correction code will adequately protect our data for factoring a 1,024-bit number even when the qubit teleportation failure rate is one percent.
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