Symmetry-based Perspectives on Hamiltonian Quantum Search Algorithms and Schrodinger's Dynamics between Orthogonal States

TL;DR

The paper analyzes why constant-Hamiltonian evolutions in a two-dimensional subspace spanned by orthogonal source and target states fail to achieve time-optimal transitions, linking this limitation to symmetry and geodesic constraints in quantum state space. It complements geometry-based arguments with energy-based reasoning to show that, for orthogonal states, a time-independent Hamiltonian cannot produce a suboptimal path within the two-dimensional subspace, while time-dependent Hamiltonians or higher-dimensional extensions can realize non-geodesic trajectories. The work also connects these insights to analog quantum search, showing that symmetry-induced level crossings under stationary schemes underlie failure modes, and suggests routes to avoid failures via symmetry breaking (couplings) or by expanding to larger subspaces or time-dependent protocols. Overall, the results illuminate the deep role of symmetry in quantum control and search, and provide concrete constructions and conditions under which time-optimality can or cannot be achieved in both stationary and nonstationary settings. The findings have implications for designing robust analog quantum search protocols and for understanding fundamental speed limits in controlled quantum evolutions.

Abstract

It is known that the continuous-time variant of Grover's search algorithm is characterized by quantum search frameworks that are governed by stationary Hamiltonians, which result in search trajectories confined to the two-dimensional subspace of the complete Hilbert space formed by the source and target states. Specifically, the search approach is ineffective when the source and target states are orthogonal. In this paper, we employ normalization, orthogonality, and energy limitations to demonstrate that it is unfeasible to breach time-optimality between orthogonal states with constant Hamiltonians when the evolution is limited to the two-dimensional space spanned by the initial and final states. Deviations from time-optimality for unitary evolutions between orthogonal states can only occur with time-dependent Hamiltonian evolutions or, alternatively, with constant Hamiltonian evolutions in higher-dimensional subspaces of the entire Hilbert space. Ultimately, we employ our quantitative analysis to provide meaningful insights regarding the relationship between time-optimal evolutions and analog quantum search methods. We determine that the challenge of transitioning between orthogonal states with a constant Hamiltonian in a sub-optimal time is closely linked to the shortcomings of analog quantum search when the source and target states are orthogonal and not interconnected by the search Hamiltonian. In both scenarios, the fundamental cause of the failure lies in the existence of an inherent symmetry within the system.

Figures (3)

• Figure 1: In (a), there is sketch of the geometry of Bloch vectors for a time optimal evolution (with $\Lambda E=\Delta E_{\max}$) between orthogonal states $\left\vert A\right\rangle$ and $\left\vert B\right\rangle$ specified by a constant Hamiltonian. In this case, we have $\mathbf{a\cdot e}_{-}=0=\mathbf{a\cdot e}_{+}$ and $\mathbf{b\cdot e}_{-}=0=\mathbf{b\cdot e}_{+}$. Clearly, $\mathbf{a\cdot b=}-1\mathbf{=e}_{-}\cdot\mathbf{e}_{+}$. In (b), there is sketch of the geometry of Bloch vectors for a time suboptimal evolution (with $\Delta E\left( t\right) <\Delta E_{\max}$) between orthogonal states $\left\vert A\right\rangle$ and $\left\vert B\right\rangle$ specified by a nonstationary Hamiltonian. In this case, we generally have $\mathbf{a\cdot e}_{-}\left( t\right) \neq0=\mathbf{a\cdot e}_{+}\left( t\right)$ and $\mathbf{b\cdot e}_{-}\left( t\right) =0=\mathbf{b\cdot e}_{+}\left( t\right)$. Clearly, $\mathbf{a\cdot b=}-1\mathbf{=e}_{-}\left( t\right) \cdot\mathbf{e}_{+}\left( t\right)$.
• Figure 2: In (a), there are the plots of the probabilities P$_{A}\left( t\right) \overset{\text{def}}{=}\left\vert \left\langle E_{+}(t)\left\vert A\right. \right\rangle \right\vert ^{2}$ (dashed line) and P$_{A}\left( t\right) \overset{\text{def}}{=}\left\vert \left\langle E_{-}(t)\left\vert A\right. \right\rangle \right\vert ^{2}$ (thick solid line) versus time $t$. In (b), there are the plots of the probabilities P$_{B}\left( t\right) \overset{\text{def}}{=}\left\vert \left\langle E_{+}(t)\left\vert B\right. \right\rangle \right\vert ^{2}$ (dashed line) and P$_{B}\left( t\right) \overset{\text{def}}{=}\left\vert \left\langle E_{-}(t)\left\vert B\right. \right\rangle \right\vert ^{2}$ (thick solid line) versus time $t$. In all plots, we set $\nu_{0}=\omega_{0}=1$ and $0\leq t\leq\pi/2$.
• Figure 3: In (a), there is a sketch of the two eigenvalues $\xi$ and $1-\xi$ of the time-dependent Hamiltonian H̃$\left( \xi\right) \overset{\text{def}}{=}(1-\xi)\left[ \mathbf{1+}\left\vert s\right\rangle \left\langle s\right\vert \right] +\xi\left[ \mathbf{1+}\left\vert w\right\rangle \left\langle w\right\vert \right]$ when $\left\vert s\right\rangle \overset{\text{def}}{=}\left\vert 0\right\rangle$, $\left\vert w\right\rangle \overset{\text{def}}{=}\left\vert 1\right\rangle$, $\left\vert \left\langle s\left\vert w\right. \right\rangle \right\vert =0$, and the reduced time $\xi$ being such that $0\leq\xi\leq1$. In this case, $\mathbf{1+}\left\vert s\right\rangle \left\langle s\right\vert$ and $\mathbf{1+}\left\vert w\right\rangle \left\langle w\right\vert$ are diagonal in the same basis, the eigenvalues cross and, therefore, $g_{\min}=0$. In (b), there is a sketch of the two eigenvalues $(1+\sqrt{\left( 1-\xi\right) ^{2}+\xi^{2}})/2$ and $(1-\sqrt{\left( 1-\xi\right) ^{2}+\xi^{2}})/2$ of the time-dependent Hamiltonian H̃$\left( \xi\right) \overset{\text{def}}{=}(1-\xi)\left[ \mathbf{1+}\left\vert s\right\rangle \left\langle s\right\vert \right] +\xi\left[ \mathbf{1+}\left\vert w\right\rangle \left\langle w\right\vert \right]$ when $\left\vert s\right\rangle \overset{\text{def}}{=}\left\vert 0\right\rangle$, $\left\vert w\right\rangle \overset{\text{def}}{=}\left[ \left\vert 0\right\rangle +\left\vert 1\right\rangle \right] /2$, $\left\vert \left\langle s\left\vert w\right. \right\rangle \right\vert \neq0$, and $0\leq\xi\leq1$. In this case, $\mathbf{1+}\left\vert s\right\rangle \left\langle s\right\vert$ and $\mathbf{1+}\left\vert w\right\rangle \left\langle w\right\vert$ are not diagonal in the same basis, the eigenvalues do not cross and, therefore, the minimum energy gap does not vanish since $g_{\min}\neq0$.