Coherent Quantum Schrodinger Bridge: Two-Boundary Optimal Control for Quantum Algorithm Design

Abstract

Quantum algorithms are intrinsically two-boundary processes: an input state is prepared, and an output state or subspace is selected as the computational answer. We formulate this observation as a coherent Quantum Schrödinger Bridge (QSB), a pure-state Hamiltonian counterpart of Schrödinger bridge theory in which the endpoint constraint is imposed on state vectors and the transport cost is the quadratic control action. In this setting Aharonov's two-state vector becomes the natural optimal-control pair: a forward state from the input and a backward state from the target. Pontryagin's principle then yields a universal optimal Hamiltonian whose weak value is purely imaginary in the geodesic gauge. Thus weak values are not an auxiliary interpretation; they are the local response functions that quantify the drift of the pre-selected state toward the post-selected boundary. Applying this framework to unstructured search, periodicity finding, and matrix arithmetic, we reconstruct Grover's algorithm, the quantum Fourier transform underlying Shor's algorithm, and quantum singular value transformation (QSVT). The usual circuit components -- oracles, diffusion reflections, controlled phases, and signal-processing rotations -- emerge as Lie-algebraic syntheses of the optimal weak-value drift. This perspective unifies distinct algorithmic paradigms into a single geometric principle: algorithm design is the problem of choosing computational boundary conditions and realizing the corresponding optimal flow.

The paper asks whether quantum algorithm primitives can be organized as minimum-energy unitary transport between prescribed input and output states. It defines a coherent, pure-state “Quantum Schrödinger Bridge” by minimizing $J[H]=\frac12\int_0^T\mathrm{Tr}(H(t)^2)\,dt$, with the target specified up to global phase. Using Pontryagin’s principle, it derives the candidate Hamiltonian $H^*=\frac{i}{2}(|\chi\rangle\langle\psi|-|\psi\rangle\langle\chi|)$, where forward and adjoint states evolve under the same Hamiltonian. It shows that this generator is time-independent and connects it to imaginary weak values through the response identity $\partial_g\log P_A(g)|_{g=0}=2\operatorname{Im}A_w$. Boundary-projector commutators provide a proposed route from the continuous generator to circuit operations. Three applications reinterpret existing primitives. For Grover search, the construction identifies the two-dimensional rotation direction generated by oracle and diffusion reflections, with input–target overlap $1/\sqrt N$. For the QFT, it proposes deriving controlled-phase interactions from Fourier-basis boundary phases. For QSVT, it assigns a rotation angle $\arccos p(\sigma)$ to each singular-value subspace and invokes existing QSP/QSVT synthesis theorems to connect admissible polynomial profiles to phase sequences. No new query-complexity bound or phase-synthesis algorithm is provided. The scope is explicitly restricted to coherent pure-state control, not stochastic, channel, or mixed-state Schrödinger bridges. Direct implementation of the resulting generators is generally unavailable. Unknown target boundaries, including ground states sought by variational algorithms, remain outside the exact construction; intermediate-boundary methods are proposed as future work.