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arXiv:0805.4250

Covariant versions of Margolus-Levitin Theorem

Qiao-jun Cao, Yi-Xin Chen, Jianlong Li

Published May 28, 2008• 4 citations
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Abstract

The Margolus-Levitin Theorem is a limitation on the minimal evolving time of a quantum system from its initial state to its orthogonal state. It also supplies a bound on the maximal operations or events can occur within a volume of spacetime. We compare Margolus-Levitin Theorem with other form of limitation on minimal evolving time, and present covariant versions of Margolus-Levitin Theorem in special and general relativity respectively. The relation among entropy bound, maximum information flow and computational limit are discussed, by applying the covariant versions of Margolus-Levitin Theorem.

DOI
10.48550/arxiv.0805.4250

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Europhysics Letters (EPL)•Jul 1, 2013•1 citation
The Margolus-Levitin Theorem is a limitation on the minimal evolving time of a quantum system from its initial state to its orthogonal state. It also supplies a bound on the maximal operations or events can occur within a volume of spacetime. We compare Margolus-Levitin Theorem with other form of limitation on minimal evolving time, and present covariant versions of Margolus-Levitin Theorem in special and general relativity respectively. The relation among entropy bound, maximum information flow and computational limit are discussed, by applying the covariant versions of Margolus-Levitin Theorem.
The Mandelstam-Tamm and Margolus-Levitin quantum speed limits are two well-known evolution time estimates for isolated quantum systems. These bounds are usually formulated for fully distinguishable initial and final states, but both have tight extensions to systems that evolve between states with an arbitrary fidelity. However, the foundations of these extensions differ in some essential respects. The extended Mandelstam-Tamm quantum speed limit has been proven analytically and has a clear geometric interpretation. Furthermore, which systems saturate the limit is known. The derivation of the extended Margolus-Levitin quantum speed limit, on the other hand, is based on numerical estimates. Moreover, the limit lacks a geometric interpretation, and no complete characterization of the systems reaching it exists. In this paper, we derive the extended Margolus-Levitin quantum speed limit analytically and describe the systems that saturate the limit in detail. We also provide the limit with a symplectic-geometric interpretation, which indicates that it is of a different character than most existing quantum speed limits. At the end of the paper, we analyze the maximum of the extended Mandelstam-Tamm and Margolus-Levitin quantum speed limits and derive a dual version of the extended Margolus-Levitin quantum speed limit. The maximum limit is tight regardless of the fidelity of the initial and final states. However, the conditions under which the maximum limit is saturated differ depending on whether or not the initial state and the final state are fully distinguishable. The dual limit is also tight and follows from a time reversal argument. We describe the systems that saturate the dual quantum speed limit.
The Margolus-Levitin (ML) bound establishes a fundamental limit on the speed of quantum evolution, stating that a system with mean energy ⟨H⟩\langle H \rangle⟨H⟩ above its ground state requires a minimum time τ≥πℏ/(2⟨H⟩)\tau \ge \pi\hbar/(2\langle H\rangle)τ≥πℏ/(2⟨H⟩) to evolve between orthogonal states. This preprint rigorously derives the ML bound and critically examines its applicability as a universal per-operation energy-time limit for quantum computation at the Landauer-relevant energy scale. We show that while the ML bound constrains the intrinsic dynamics of the qubit register, it does not account for the dominant energy dissipation from cryogenic control hardware. Consequently, the ML bound alone is insufficient to characterize the energy cost of a logical operation. To address this, we connect the theoretical limit to the QNFO joules-per-compute (JPCUB) benchmark program and propose a concrete normalization of computational cost. This normalization quantifies energy per logical qubit operation by incorporating both the intrinsic qubit energy and the overhead of cryogenic controllers, providing a practical metric for quantum computation efficiency. ## 1. Introduction Quantum speed limits (QSLs) bound the minimum time required for a quantum state to evolve to an orthogonal state. The Margolus-Levitin (ML) bound is a prominent QSL, relating this minimum time to the mean energy of the system. As quantum computing hardware matures, understanding the fundamental energy limits of computation becomes critical. The Landauer limit sets a lower bound on the energy dissipated by irreversible operations, while the ML bound sets a lower bound on the time required for state evolution. This preprint derives the ML bound and assesses its role as a universal per-operation energy-time bound for quantum computation. We then connect this theoretical limit to the QNFO joules-per-compute (JPCUB) benchmark program, proposing a concrete normalization for the energy cost of logical qubit operations in cr
α>(ε)=α<(ε)\boldsymbol{α_{>}(ε) = α_{<}(ε)}α>​(ε)=α<​(ε) For The Margolus-Levitin Quantum Speed Limit Bound
The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least πα(ε)/(2⟨E−E0⟩)πα(ε) / (2 \langle E-E_0 \rangle)πα(ε)/(2⟨E−E0​⟩), where ⟨E−E0⟩\langle E-E_0 \rangle⟨E−E0​⟩ is the expected energy of the system relative to the ground state of the Hamiltonian and α(ε)α(ε)α(ε) is a function of the fidelity εεε between the two state. For a long time, only a upper bound α>(ε)α_{>}(ε)α>​(ε) and lower bound α<(ε)α_{<}(ε)α<​(ε) are known although they agree up to at least seven significant figures. Lately, Hörnedal and Sönnerborn proved an analytical expression for α(ε)α(ε)α(ε), fully classified systems whose evolution times saturate the ML bound, and gave this bound a symplectic-geometric interpretation. Here I solve the same problem through an elementary proof of the ML bound. By explicitly finding all the states that saturate the ML bound, I show that α>(ε)α_{>}(ε)α>​(ε) is indeed equal to α<(ε)α_{<}(ε)α<​(ε). More importantly, I point out a numerical stability issue in computing α>(ε)α_{>}(ε)α>​(ε) and report a simple way to evaluate it efficiently and accurately.
Some quantum states can dynamically evolve to an orthogonal state after a time interval, while some states can never evolve to their orthogonal states. In this paper, we investigate three issues on the orthogonalization properties of quantum states. First, we answer the question as to under what conditions an initial state can evolve to an orthogonal state. Second, we discuss the orthogonalization rate of quantum states, i.e. , the total number of orthogonal states that the evolution can go through in unit time. Finally, we show that the Margolus-Levitin bound for orthogonalization time remains tight for superpositions of more than two eigenstates.