Quantum Speed Limits for Implementation of Unitary Transformations

TL;DR

This work derives quantum speed limits for implementing unitary transformations in finite-dimensional quantum systems, with bounds that depend only on the trace of the target unitary and the energy statistics of the generator, and are invariant to global energy shifts. The main result provides MT- and ML-like bounds for unitary gates via $T \ge \max\left( \frac{\pi}{2E}\left(1 - \frac{|tr(U)|}{N} \sqrt{1 + \frac{4}{\pi^2}}\right), \frac{1}{\Delta E}\sqrt{1 - \frac{|tr(U)|^2}{N^2}} \right)$, plus a corollary in terms of the spectrum width $\delta E$. The authors prove the bounds using simple trigonometric and cosine inequalities and express the results entirely through $|tr(U)|$ to avoid phase ambiguities from taking logarithms of $U$. They apply the bounds to transformations of MUBs, permutation operators, and the Grover operator, and verify tightness in the qubit case with an exact energy-time relation, while providing numerical insights for qutrits. Overall, the results offer a practical tool for designing quantum circuits by specifying speed limits for implementing arbitrary unitaries in terms of easily computable spectral quantities.

Abstract

Quantum speed limits are the boundaries that define how quickly one quantum state can transform into another. Instead of focusing on the transformation between pairs of states, we provide bounds on the speed limit of quantum evolution by unitary operators in arbitrary dimensions. These do not depend on the initial and final state but depend only on the trace of the unitary operator that is to be implemented and the gross characteristics (average and variance) of the energy spectrum of the Hamiltonian which generates this unitary evolution. The bounds that we find can be thought of as the generalization of the Mandelstam-Tamm (TM) and the Margolus-Levitin (ML) bound for state transformations to implementations of unitary operators. We will discuss the application of these bounds in several classes of transformations that are of interest in quantum information processing.

Figures (5)

• Figure 1: (Color online) The exact time versus $|\tr(U)|$ for enacting a unitary operator on a qubit system (solid line), compared with the bound (\ref{['b1']}) (the dashed line). For qubits, $\Delta E=E=\frac{E_2-E_1}{2}$ and hence both bounds in (\ref{['b1']}) can be expressed as $ET$ versus $|\tr(U)|$.
• Figure 2: The basis $B_0=\{|0\rangle,|1\rangle\}$ can be transformed to a basis in the set of all MUB's (shaded region). With a given energy, transforming to the basis $B_1^*=\{ |e_0\rangle,|e_1\rangle\}$ as given in (\ref{['b1*']}) needs the shortest amount of time.
• Figure 3: The manifold of MUB bases for qutrits has two disjoint parts. With a given amount of energy, the basis $B_0=\{|0\rangle,|1\rangle, |2\rangle\}$ can be transformed in the shortest time to a MUB basis $B_1 \in {\cal M}_I$ or to a MUB basis $B_2\in {\cal M}_{II}$. The energy-time bound for the two sets are different.
• Figure 4: The exact energy-time relation for the unitary $U_1$ as a function of $y$ for various values of $x$(continuous line) versus the time-energy bound (\ref{['ineq1']}) (dashed line).
• Figure 5: The exact energy-time relation for the unitary $U_2$ as a function of $y$ for various values of $x$(continuous line) versus the time-energy bound (\ref{['ineq1']}) (dashed line).