A mathematical model for a universal digital quantum computer with an application to the Grover-Rudolph algorithm

TL;DR

The paper introduces an algebraic probability framework for gate-based quantum computation, treating quantum circuits as unitary actions on density operators and unifying unitary evolution with measurements. It proves that elementary quantum gates form a universal dictionary for $U(N)$, enabling any unitary to be constructed from a finite sequence of embeddings of $U(2)$ into $U(N)$. The authors apply this framework to encode probability distributions in quantum circuits, generalizing the Grover–Rudolph approach by providing a constructive decomposition that yields a circuit of length $N-1$ whose outcome probabilities approximate a given density via $\mathbb{P}_{\mathcal{N}_{\rho_0}(\mathsf{U})}(\mathsf{A}=k)=\int_{k/2^n}^{(k+1)/2^n}\varrho(x)\,dx$. Numerical simulations using Qiskit validate the method’s feasibility and accuracy for encoding distributions, with potential applications to quantum machine learning and circuit optimization. The work offers a principled route to quantum circuit synthesis within an algebraic-probability setting, bridging classical and quantum probabilistic formalisms.

Abstract

In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their role in implementing unitary transformations. A key result demonstrates that every unitary matrix in \(\mathrm{U}(N)\) can be expressed as a product of elementary quantum gates, leading to the concept of a universal dictionary for quantum computation. We apply this framework to the construction of quantum circuits that encode probability distributions, focusing on the Grover-Rudolph algorithm. By leveraging controlled quantum gates and rotation matrices, we design a quantum circuit that approximates a given probability density function. Numerical simulations, conducted using Qiskit, confirm the theoretical predictions and validate the effectiveness of our approach. These results provide a rigorous foundation for quantum circuit synthesis within an algebraic probability framework and offer new insights into the encoding of probability distributions in quantum algorithms. Potential applications include quantum machine learning, circuit optimization, and experimental implementations on real quantum hardware.

Figures (10)

• Figure 3.1: Wire diagram for a $n$-qubit universal digital quantum computer.
• Figure 3.2: Wire diagram of the elementary quantum gate $w_j^{(n)}(\mathsf{U}) = \mathsf{I}_2^{\otimes (j-1)} \otimes \mathsf{U} \otimes \mathsf{I}_2^{\otimes (n-j)}.$
• Figure 3.3: Wire diagram of the product of elementary quantum gates $w_1^{(n)}(\mathsf{V})w_j^{(n)}(\mathsf{U}) = \mathsf{V} \otimes \mathsf{I}_2^{\otimes (j-2)} \otimes \mathsf{U} \otimes \mathsf{I}_2^{\otimes (n-j)}.$
• Figure 4.1: Wire diagram for the CNOT elementary quantum gate $\mathsf{C}\mathsf{C}_{0}^{(1)}\mathsf{X}.$
• Figure 4.2: The density function $\varrho(x)$.

Theorems & Definitions (44)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3: Spectral Theorem
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Theorem 2.9
• proof