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n-Qubit Operations on Sphere and Queueing Scaling Limits for Programmable Quantum Computer

Wanyang Dai

TL;DR

This work addresses how to operationalize programmable quantum computation by formulating $n$-qubit operations on the $(n+1)$-sphere $S^{n+1}$ and by deriving scaling limits for quantum queueing systems driven by batch quantum random walks on $S^{n+1}$. The main methodological contributions are the rigorous construction of synchronized $n$-qubit operations that preserve normalization on $S^{n+1}$ and the establishment of reflecting Gaussian random-field (RGRF) diffusion limits under two heavy-traffic regimes, including a fixed-qubit-number regime and a variable-qubit-number regime, with drift $ heta t$ and Brownian-driven inputs. The key results show that, under appropriate heavy-traffic scalings, the total workload converges to a RGRF on $S^{n+1}$ (or its infinite-dimensional limit $S^{ fty}$), enabling performance analysis and design guidance for quantum buffers, CPUs, and quantum-channel interactions. These findings have practical implications for selecting an appropriate qubit count and for performance planning in photon- or other-tech programmable quantum computers, by linking quantum-state operations on spheres to queueing-theoretic scaling laws.

Abstract

We study n-qubit operation rules on (n+1)-sphere with the target to help developing a (photon or other technique) based programmable quantum computer. In the meanwhile, we derive the scaling limits (called reflecting Gaussian random fields on a (n+1)-sphere) for n-qubit quantum computer based queueing systems under two different heavy traffic regimes. The queueing systems are with multiple classes of users and batch quantum random walks over the $(n+1)$-sphere as arrival inputs. In the first regime, the qubit number $n$ is fixed and the scaling is in terms of both time and space. Under this regime, performance modeling during deriving the scaling limit in terms of balancing the arrival and service rates under first-in first-out and work conserving service policy is conducted. In the second regime, besides the time and space scaling parameters, the qubit number $n$ itself is also considered as a varying scaling parameter with the additional aim to find a suitable number of qubits for the design of a quantum computer. This regime is in contrast to the well-known Halfin-Whitt regime.

n-Qubit Operations on Sphere and Queueing Scaling Limits for Programmable Quantum Computer

TL;DR

This work addresses how to operationalize programmable quantum computation by formulating -qubit operations on the -sphere and by deriving scaling limits for quantum queueing systems driven by batch quantum random walks on . The main methodological contributions are the rigorous construction of synchronized -qubit operations that preserve normalization on and the establishment of reflecting Gaussian random-field (RGRF) diffusion limits under two heavy-traffic regimes, including a fixed-qubit-number regime and a variable-qubit-number regime, with drift and Brownian-driven inputs. The key results show that, under appropriate heavy-traffic scalings, the total workload converges to a RGRF on (or its infinite-dimensional limit ), enabling performance analysis and design guidance for quantum buffers, CPUs, and quantum-channel interactions. These findings have practical implications for selecting an appropriate qubit count and for performance planning in photon- or other-tech programmable quantum computers, by linking quantum-state operations on spheres to queueing-theoretic scaling laws.

Abstract

We study n-qubit operation rules on (n+1)-sphere with the target to help developing a (photon or other technique) based programmable quantum computer. In the meanwhile, we derive the scaling limits (called reflecting Gaussian random fields on a (n+1)-sphere) for n-qubit quantum computer based queueing systems under two different heavy traffic regimes. The queueing systems are with multiple classes of users and batch quantum random walks over the -sphere as arrival inputs. In the first regime, the qubit number is fixed and the scaling is in terms of both time and space. Under this regime, performance modeling during deriving the scaling limit in terms of balancing the arrival and service rates under first-in first-out and work conserving service policy is conducted. In the second regime, besides the time and space scaling parameters, the qubit number itself is also considered as a varying scaling parameter with the additional aim to find a suitable number of qubits for the design of a quantum computer. This regime is in contrast to the well-known Halfin-Whitt regime.

Paper Structure

This paper contains 14 sections, 3 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $n$-Qubit Operations on $(n+1)$-Sphere
  3. The Operations
  4. Illustration and Examples
  5. Queueing Scaling Limits
  6. Preliminaries and Performance Measures
  7. Heavy Traffic Limit Theorems
  8. The Case Corresponding to Fixed Qubit Number
  9. The Case Corresponding to Variable Qubit Number
  10. Proofs of Heavy Traffic Limit Theorems
  11. Appendix: Proof of Proposition \ref{['fgshajksda']}
  12. Proof of Part I.
  13. Proof of Part II
  14. Conclusion

Key Result

Proposition 2.1

The statement of this proposition consists of the following two parts:

Figures (4)

  • Figure 1: A quantum buffer queueing storage, quantum computer central processing unit (CPU) with $n$-qubit operations $|\Upsilon\rangle(\Phi,\Psi)$ over unit $(n+1)$-sphere as introduced in (\ref{['e:jkdsalp']}), and quantum channel measurement interactive system with channel function $G(\Phi,\Psi)$ derived in (\ref{['e:channelphipsi']}).
  • Figure 2: Single qubit representation, quantum random walk on $S^{2}$, and converging sets.
  • Figure 3: Single qubit representation on $S^{2}$
  • Figure 4: An example of the interaction between the quantum channel model in Dai dai:quacom and a photon based quantum computer core, where the inner photon detecting and measurement part is adapted from Zhong et al.zho:quacom.

Theorems & Definitions (5)

  • Definition 1.1
  • Proposition 2.1
  • Definition 3.1
  • Theorem 3.1
  • Theorem 3.2