Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem

• URL: http://arxiv.org/abs/2406.04846v2
• Date: Sun, 30 Jun 2024 07:24:25 GMT
• Title: Efficient Fault-Tolerant Single Qubit Gate Approximation And Universal Quantum Computation Without Using The Solovay-Kitaev Theorem
• Authors: H. F. Chau,
• Abstract summary: A recent improvement of the Solovay-Kitaev theorem implies that to approximate any single-qubit gate to an accuracy of $epsilon > 0$ requires $textO(logc[1/epsilon)$ quantum gates with $c > 1.44042$. Here I give a partial answer to this question by showing that this is possible using $textO(log[/epsilon] loglog[/epsilon] cdots)$ FT gates chosen from a finite set depending on the value of $
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Arbitrarily accurate fault-tolerant (FT) universal quantum computation can be carried out using the Clifford gates Z, S, CNOT plus the non-Clifford T gate. Moreover, a recent improvement of the Solovay-Kitaev theorem by Kuperberg implies that to approximate any single-qubit gate to an accuracy of $\epsilon > 0$ requires $\text{O}(\log^c[1/\epsilon])$ quantum gates with $c > 1.44042$. Can one do better? That was the question asked by Nielsen and Chuang in their quantum computation textbook. Specifically, they posted a challenge to efficiently approximate single-qubit gate, fault-tolerantly or otherwise, using $\Omega(\log[1/\epsilon])$ gates chosen from a finite set. Here I give a partial answer to this question by showing that this is possible using $\text{O}(\log[1/\epsilon] \log\log[1/\epsilon] \log\log\log[1/\epsilon] \cdots)$ FT gates chosen from a finite set depending on the value of $\epsilon$. The key idea is to construct an approximation of any phase gate in a FT way by recursion to any given accuracy $\epsilon > 0$. This method is straightforward to implement, easy to understand, and interestingly does not involve the Solovay-Kitaev theorem.

Related papers

• Constant-depth circuits for Uniformly Controlled Gates and Boolean functions with application to quantum memory circuits [42.979881926959486]
  We propose two types of constant-depth constructions for implementing Uniformly Controlled Gates. We obtain constant-depth circuits for the quantum counterparts of read-only and read-write memory devices.
  arXiv  Detail & Related papers  (2023-08-16T17:54:56Z)
• Spacetime-Efficient Low-Depth Quantum State Preparation with Applications [93.56766264306764]
  We show that a novel deterministic method for preparing arbitrary quantum states requires fewer quantum resources than previous methods. We highlight several applications where this ability would be useful, including quantum machine learning, Hamiltonian simulation, and solving linear systems of equations.
  arXiv  Detail & Related papers  (2023-03-03T18:23:20Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• Shorter quantum circuits via single-qubit gate approximation [1.2463824156241843]
  We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set. We show that taking probabilistic mixtures of channels to solve (arXiv:1409.3552) and magnitude approximation problems saves factor of two in approximation costs.
  arXiv  Detail & Related papers  (2022-03-18T17:14:35Z)
• A lower bound on the space overhead of fault-tolerant quantum computation [51.723084600243716]
  The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation. We prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude noise.
  arXiv  Detail & Related papers  (2022-01-31T22:19:49Z)
• Random quantum circuits transform local noise into global white noise [118.18170052022323]
  We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. For local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_textnoisy$ of a generic noisy circuit instance shrink exponentially. If the noise is incoherent, the output distribution approaches the uniform distribution $p_textunif$ at precisely the same rate.
  arXiv  Detail & Related papers  (2021-11-29T19:26:28Z)
• T-count and T-depth of any multi-qubit unitary [1.933681537640272]
  We design provable algorithm to determine T-count of any $n$-qubit ($ngeq 1$) unitary $W$ of size $2ntimes 2n$, over the Clifford+T gate set. Our algorithm can also be used to determine the (minimum possible) T-depth of any multi-qubit unitary.
  arXiv  Detail & Related papers  (2021-10-19T22:16:00Z)
• Quantum supremacy and hardness of estimating output probabilities of quantum circuits [0.0]
  We prove under the theoretical complexity of the non-concentration hierarchy that approximating the output probabilities to within $2-Omega(nlogn)$ is hard. We show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including trivial circuit with identity gates.
  arXiv  Detail & Related papers  (2021-02-03T09:20:32Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• Trading T gates for dirty qubits in state preparation and unitary synthesis [0.0]
  We present a quantum algorithm for preparing any dimension-$N$ pure quantum state specified by a list of $N$ classical numbers. Our scheme uses $mathcalO(fracNlambda+lambdalogfracNepsilonlogfraclogNepsilon)$ to reduce the T-gate cost to $mathcalO(fracNlambda+lambdalogfracNepsilonlogfraclogNepsilon)$
  arXiv  Detail & Related papers  (2018-12-03T18:24:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.