[Paper Review] The sum-over-histories formulation of quantum computing

Ben Rudiak-Gould|ArXiv.org|Jul 21, 2006

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper introduces the sum-over-histories formalism as a complementary framework to the standard state-vector approach in quantum computing, using path integral principles to model quantum circuits. It demonstrates mathematical equivalence to the canonical formalism, showing that quantum amplitude calculations can be framed as sums over all possible computational histories, offering new insights into complexity and potential advantages in circuit simulation and quantum language design.

ABSTRACT

Since Deutsch (1985), quantum computers have been modeled exclusively in the language of state vectors and the Schroedinger equation. We present a complementary view of quantum circuits inspired by the path integral formalism of quantum mechanics, and examine its application to some simple textbook problems.

Motivation & Objective

• To present an alternative formalism for quantum computing based on the path integral (sum-over-histories) approach, rather than the standard state-vector and Schrödinger equation framework.
• To bridge the gap between quantum field theory techniques and quantum circuit design by adapting path integral methods to discrete quantum circuits.
• To explore whether the sum-over-histories formalism offers conceptual or computational advantages in understanding quantum algorithms, complexity, and circuit simulation.
• To investigate its potential utility in quantum programming language design, particularly for topological or non-sequential circuit models.
• To provide an intuitive, alternative perspective that may aid in teaching and conceptual understanding of quantum computation.

Proposed method

• Discretize the continuous path integral formalism by restricting the system to 2^n classical states and dividing time into discrete intervals of length δt.
• Represent each quantum history as a sequence of computational basis states over time, forming a discrete sum over all possible state trajectories.
• Define a discretized Lagrangian L(φ_t, φ_{t+δt}, t) based on transitions between consecutive states, which determines the phase contribution of each history.
• Express the total transition amplitude as a product of complex weights B(φ_t, φ_{t+δt}, t) = exp(i/ħ × L(φ_t, φ_{t+δt}, t)) summed over all histories.
• Establish mathematical equivalence between the canonical unitary evolution and the sum-over-histories formulation by showing both yield the same amplitude for a given input-output transition.
• Apply the formalism to textbook quantum circuits to illustrate its conceptual clarity and computational feasibility.

Experimental results

Research questions

• RQ1 Can the sum-over-histories formalism be effectively applied to model standard quantum circuits, providing an alternative to the canonical state-vector approach?
• RQ2 How does the sum-over-histories formalism compare to the canonical formalism in terms of computational complexity and space efficiency for circuit simulation?
• RQ3 What insights does the path integral perspective offer into the structure of quantum algorithms and the relationship between classical and quantum computation?
• RQ4 Can this formalism support novel quantum programming language designs, particularly for topological or non-sequential quantum circuits?
• RQ5 Does the sum-over-histories approach simplify or clarify foundational concepts in quantum complexity, such as the containment of BQP in PSPACE?

Key findings

• The sum-over-histories formalism is mathematically equivalent to the canonical unitary evolution formalism, with both yielding identical transition amplitudes for any quantum circuit.
• The formalism makes the containment of BQP in PSPACE conceptually transparent, as the sum over histories can be computed in polynomial space.
• Naïve simulation using the sum-over-histories approach requires significantly less space than state-vector simulation, especially for large systems.
• The approach provides an intuitive framework for understanding quantum circuits as collections of histories, where classical gates effectively filter or reject most histories.
• The formalism may aid in education by offering a dual perspective that helps learners grasp quantum mechanics beyond the abstract state vector interpretation.
• While not expected to yield new polynomial-time simulation algorithms, the formalism offers conceptual advantages in complexity theory and may inspire new abstractions in quantum language design.