Almost Optimal Synthesis of Reversible Function in Qudit Model

• URL: http://arxiv.org/abs/2501.05237v1
• Date: Thu, 09 Jan 2025 13:44:14 GMT
• Title: Almost Optimal Synthesis of Reversible Function in Qudit Model
• Authors: Buji Xu, Junhong Nie, Xiaoming Sun,
• Abstract summary: We propose a method to synthesize even permutations in $A_dn$ using $Theta(d)$(n - 1)$-qudit sub-circuits. We also introduce a technique for reversible functions employing $Oleft( n dn right)$ gates and only a single ancilla.
• Score: 5.9453200974654195
• License:
• Abstract: Quantum oracles are widely adopted in problems, like query oracle in Grover's algorithm, cipher in quantum cryptanalytic and data encoder in quantum machine learning. Notably, the bit-flip oracle, capable of flipping the state based on a given classical function, emerges as a fundamental component in the design and construction of quantum algorithms. Devising methods to optimally implement the bit-flip oracle essentially translates to the efficient synthesis of reversible functions. Prior research has primarily focused on the qubit model, leaving the higher dimensional systems, i.e. qudit model, largely unexplored. By allowing more than two computational bases, qudit model can fully utilize the multi-level nature of the underlying physical mechanism. We propose a method to synthesize even permutations in $A_{d^{n}}$ using $\Theta(d)$ $(n - 1)$-qudit sub-circuits, which achieve asymptotic optimality in the count of sub-circuits. Moreover, we introduce a technique for synthesizing reversible functions employing $O\left( n d^{n} \right)$ gates and only a single ancilla. This is asymptotically tight in terms of $d$ and asymptotically almost tight in terms of $n$.

Related papers

• Quantum Approximate $k$-Minimum Finding [2.810947654192424]
  We propose an optimal quantum $k$-minimum finding algorithm that works with approximate values for all $k geq 1$. We present efficient quantum algorithms for identifying the $k$ smallest expectation values among multiple observables and for determining the $k$ lowest ground state energies of a Hamiltonian.
  arXiv  Detail & Related papers  (2024-12-21T11:21:15Z)
• Multi-strategy Based Quantum Cost Reduction of Quantum Boolean Circuits [0.4999814847776098]
  The construction of quantum computers is based on the synthesis of low-cost quantum circuits. This paper proposes two algorithms to construct a quantum circuit for any Boolean function expressed in a Positive Polarity Reed-Muller $PPRM$ expansion.
  arXiv  Detail & Related papers  (2024-07-05T19:25:46Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• A one-query lower bound for unitary synthesis and breaking quantum cryptography [7.705803563459633]
  The Unitary Synthesis Problem asks whether any $n$qubit unitary $U$ can be implemented by an efficient quantum $A$ augmented with an oracle that computes an arbitrary Boolean function $f$. In this work, we prove unitary synthesis as an efficient challenger-ad game, which enables proving lower bounds by analyzing the maximum success probability of an adversary $Af$.
  arXiv  Detail & Related papers  (2023-10-13T05:39:42Z)
• Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications [0.0]
  Quantum algorithms manipulate the amplitudes of quantum states to find solutions to computational problems. We present a framework for applying a general class of non-linear functions to the amplitudes of quantum states. Our work provides an important and efficient building block with potentially numerous applications in areas such as optimization, state preparation, quantum chemistry, and machine learning.
  arXiv  Detail & Related papers  (2023-09-18T14:57:21Z)
• Efficient quantum linear solver algorithm with detailed running costs [0.0]
  We introduce a quantum linear solver algorithm combining ideasdiabatic quantum computing with filtering techniques based on quantum signal processing. Our protocol reduces the cost of quantum linear solvers over state-of-the-art close to an order of magnitude for early implementations.
  arXiv  Detail & Related papers  (2023-05-19T00:07:32Z)
• A single $T$-gate makes distribution learning hard [56.045224655472865]
  This work provides an extensive characterization of the learnability of the output distributions of local quantum circuits. We show that for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth $d=omega(log(n))$ Clifford circuits is hard.
  arXiv  Detail & Related papers  (2022-07-07T08:04:15Z)
• 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)
• Resource Optimisation of Coherently Controlled Quantum Computations with the PBS-calculus [55.2480439325792]
  Coherent control of quantum computations can be used to improve some quantum protocols and algorithms. We refine the PBS-calculus, a graphical language for coherent control inspired by quantum optics.
  arXiv  Detail & Related papers  (2022-02-10T18:59:52Z)
• Preparation of excited states for nuclear dynamics on a quantum computer [117.44028458220427]
  We study two different methods to prepare excited states on a quantum computer. We benchmark these techniques on emulated and real quantum devices. These findings show that quantum techniques designed to achieve good scaling on fault tolerant devices might also provide practical benefits on devices with limited connectivity and gate fidelity.
  arXiv  Detail & Related papers  (2020-09-28T17:21:25Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

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.