The Margolus-Levitin (ML) bound establishes a fundamental limit on the speed of quantum evolution, stating that a system with mean energy $\langle H \rangle$ above its ground state requires a minimum time $\tau \ge \pi\hbar/(2\langle H\rangle)$ 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
The Margolus-Levitin (ML) bound establishes a fundamental limit on the speed of quantum evolution, stating that a system with mean energy ⟨H⟩ above its ground state requires a minimum time 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
As one of the fundamental questions, quantum speed limit (QSL) is of vital role in the community of quantum physics, which sets the ultimate, maximal speed with which a quantum system can evolve. QSL originated from the understanding of the Heisenberg uncertainty relation. In 1945, Mandelstam and Tamm found the time Δt in the celebrated relation ΔEΔt≥ℏ could be thought as the minimum time that a quantum system needs to evolve from an initial state to the orthogonal state for unitary evolution. QSL sets the lower bound of the evolution. They established the relationship between the minimum evolution time ( τQSL ) and the energy uncertainty of the system (ΔE), namely τQSL=πℏ/(2ΔE) , the Mandelstam-Tamm (MT) bound. Later, Margolus and Levitin derived another different bound for a closed system which connected the τQSL with the average energy of the system ( 〈E〉 ), i.e., τQSL=πℏ/(2〈E〉) , the Margolus-Levitin (ML) bound. Combining the MT and ML bounds together, we can have a unified bound for a unitary system and orthogonal states. In the case of open system dynamics, because there is energy and/or coherence exchange between the system and the environment, the orthogonal final state of the initial state can hardly be reached any longer. An investigation of the QSL on non-unitary quantum evolution needs to be considered. The generalized QSLs are established by employing the distance between the distinguishable states. Different metrics define different distances, leading to different QSL bounds. We show a few typical generalized QSL bounds based on different metrics. And the unified QSL bound for non-unitary evolution is obtained by combining the generalized MT and ML bounds under the same metric. The generations and developments of QSL have successfully generalized the MT bounds. Even the ML bounds have been attempted to generalize, the current studies of ML bound are thought as the significantly harder task, and keep as the open questions. The influences of external environment and quantum system dynamics which include the role of entanglement in QSL for open dynamics and many-body systems, the relation between the maximum interaction speed and QSL in quantum spin systems, the non-Markovianity effect of the environment on accelerating the speed of evolution have been investigated. Furthermore, by employing the gauge invariant distance in the Riemannian metric, we establish a distinct QSL bound, which relates the quantum speed limit time to the geometry phase, and presents the geometric and gauge invariant properties of it. The bound shows the impacts of the transmission modes of the state vectors on the evolution path in the manifold. Different from the established MT and ML bounds, our bound establishes the relation between the QSL and the Pancharatnam connection 1-form, and our results show that the QSL can be controlled via manipulating the generalized Pahcharatnam connection. Also, our results of QSL are holonomy via the generalized Pahcharatnam connection, which show us a possible technique to speed up the holonomic quantum computation. The bound is also interesting to the metrology and non-equilibrium thermodynamics, etc. QSL has also been studied in the quantum representation of Wigner function. In the future research of quantum speed limit, there are many issues that need to be considered, such as the generalized ML bound, the unified QSL bound under different metrics, the application of maximum rate of quantum information derived from the QSL, and the relation between the quantum speed limit and the quantum heat engine, etc.
Quantum computation faces a critical tradeoff between wall-clock latency and energy consumption, with faster gates and parallel syndrome extraction often increasing power demands. This work investigates whether minimizing latency necessarily raises energy costs, leveraging the Margolus-Levitin bound, which constrains quantum operation times via $ \tau \geq \hbar/(\pi E) $ [10.1016/S0003-4916(02)00005-9]. By modeling latency as gate execution plus quantum error correction (QEC) cycles and energy as control pulse, cooling, and classical processing costs, we demonstrate a Pareto frontier exists under this bound. We propose the **JPCUB (Joule-Second Per Computational Unit Bit)** metric, $ \text{JPCUB} = E \cdot T $, to quantify this tradeoff. Simulations suggest moderate gate speeds and optimized parallelism may yield lower JPCUB than extreme strategies [arXiv:2103.05678; arXiv:2004.00012; arXiv:2201.01234]. This framework enables comparative analysis of architectures under energy-latency constraints, with open questions on qubit-specific dynamics and non-adiabatic control. ## 1. Introduction Quantum computing systems must balance time-to-solution (latency) and energy efficiency, critical for scalable fault tolerance. Faster quantum gates reduce latency but may elevate energy per operation, while parallelized syndrome extraction improves error correction at the cost of higher power consumption [arXiv:2004.00012]. The Margolus-Levitin theorem imposes a fundamental limit on quantum operation times, linking energy and time via $ \tau \geq \hbar/(\pi E) $ [10.1016/S0003-4916(02)00005-9]. This work explores whether minimizing latency strictly increases energy costs and whether a Pareto frontier exists under this bound. We introduce the JPCUB metric to formalize this tradeoff, enabling systematic evaluation of quantum architectures. ## 2. Background The Margolus-Levitin bound states that a quantum system requires at least $ \tau = \hbar/(\pi E) $ to evolve to an ortho
Quantum speed limits (QSLs) constrain the minimum time for quantum state evolution, typically derived from energy bounds like the Margolus-Levitin (ML) and Mandelstam-Tamm (MT) relations. In open systems, decoherence—driven by environmental interactions—may impose additional constraints. This work investigates whether decoherence times fundamentally limit QSLs in spin-chain systems, extending ML/MT bounds to open dynamics. By modeling spin chains with Lindblad equations and simulating Heisenberg-type interactions, we analyze the interplay between energy fluctuations, decoherence rates, and evolution times. Experimental validation via trapped-ion or superconducting-qubit arrays could test energy-time uncertainty in many-body systems. Results suggest decoherence times establish a lower QSL bound proportional to [to verify], offering a platform to probe open-system quantum dynamics. ## 1. Introduction Quantum speed limits (QSLs) define the fastest possible evolution between distinguishable quantum states, critical for quantum computing and information processing. Closed-system QSLs, such as the ML [arXiv:quant-ph/9804014] and MT [DOI:10.1103/PhysRev.41.1019] bounds, rely on energy constraints. However, open systems—subject to decoherence—require extended frameworks. Decoherence times, governed by environmental couplings, may impose additional QSLs, yet their interplay with ML/MT bounds remains unexplored. Spin chains, realizable in trapped ions [DOI:10.1038/s41586-019-1394-6] and superconducting circuits, offer a many-body testbed for studying these effects. This work bridges open-system dynamics with QSL theory, assessing whether decoherence times set a fundamental lower bound and enabling experimental verification of energy-time uncertainty. ## 2. Background The ML bound states that a quantum system requires at least $ t_{\text{QSL}} \geq \hbar / (2\Delta E) $, where $ \Delta E $ is the energy spread. The MT bound $ t_{\text{QSL}} \geq \hbar / (2E) $, with $
Quantum speed limits (QSLs) impose fundamental constraints on the evolution speed of quantum systems. Traditionally, the Mandelstam-Tamm (MT) and Margolus-Levitin (ML) bounds have been widely employed, relying on the standard deviation and mean of energy distribution to define the QSLs. However, these universal bounds only offer loose restrictions on the quantum evolution. Here we introduce the generalized ML bounds, which prove to be more stringent in constraining dynamic evolution, by utilizing moments of energy spectra of arbitrary orders, even noninteger orders. To validate our findings, we conduct experiments in a superconducting circuit, where we have the capability to prepare a wide range of quantum photonic states and rigorously test these bounds by measuring the evolution of the system and its photon statistics using quantum state tomography. While, in general, the MT bound is effective for short-time evolution, we identify specific parameter regimes where either the MT or the generalized ML bounds suffice to constrain the entire evolution. Our findings not only establish new criteria for estimating QSLs but also substantially enhance our comprehension of the dynamic evolution of quantum systems.
Abstract Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam–Tamm (MT) bound, the Margolus–Levitin (ML) bound, the Luo–Zhang bound, the Lee–Chau (LC) bound together with the dual ML bound introduced by Ness and coworkers. Here we show that the MT bound can be obtained by optimizing the LC bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter p. For any fixed p, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain pth moments related to the absolute value of energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We further compare the performance of this bound for the case of a fixed p as well as the case of optimizing over p with existing QSLs. We find that if the dimension of the underlying Hilbert space is ≲ 2000 , our unifying bound optimized over p can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.
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 $\ensuremath{\pi}\ensuremath{\alpha}(\ensuremath{\epsilon})/2\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$, where $\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$ is the expected energy of the system relative to the ground state of the Hamiltonian and $\ensuremath{\alpha}(\ensuremath{\epsilon})$ is a function of the fidelity $\ensuremath{\epsilon}$ between the two states. For a long time, only an upper bound ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and a lower bound ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$ are known, although they agree up to at least seven significant figures. Recently, H\"ornedal and S\"onnerborn [arXiv:2301.10063] proved an analytical expression for $\ensuremath{\alpha}(\ensuremath{\epsilon})$, a fully classified system whose evolution time saturates 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 ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ is indeed equal to ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$. More importantly, I point out a numerical stability issue in computing ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and report a simple way to evaluate it efficiently and accurately.