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arXiv:1701.01175 · Physical Review A

Fast quantum computation at arbitrarily low energy

Stephen P. Jordan

Published Mar 6, 2017• 40 citations
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Abstract

One version of the energy-time uncertainty principle states that the minimum time ${T}_{\ensuremath{\perp}}$ for a quantum system to evolve from a given state to any orthogonal state is $h/(4\mathrm{\ensuremath{\Delta}}E)$, where $\mathrm{\ensuremath{\Delta}}E$ is the energy uncertainty. A related bound called the Margolus-Levitin theorem states that ${T}_{\ensuremath{\perp}}\ensuremath{\ge}h/(2\ensuremath{\langle}E\ensuremath{\rangle})$, where $\ensuremath{\langle}E\ensuremath{\rangle}$ is the expectation value of energy and the ground energy is taken to be zero. Many subsequent works have interpreted ${T}_{\ensuremath{\perp}}$ as defining a minimal time for an elementary computational operation and correspondingly a fundamental limit on clock speed determined by a system's energy. Here we present local time-independent Hamiltonians in which computational clock speed becomes arbitrarily large relative to $\ensuremath{\langle}E\ensuremath{\rangle}$ and $\mathrm{\ensuremath{\Delta}}E$ as the number of computational steps goes to infinity. We argue that energy considerations alone are not sufficient to obtain an upper bound on computational speed, and that additional physical assumptions such as limits to information density and information transmission speed are necessary to obtain such a bound.

DOI
10.1103/physreva.95.032305

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One version of the energy-time uncertainty principle states that the minimum time T\ensuremath⊥{T}_{\ensuremath{\perp}}T\ensuremath⊥​ for a quantum system to evolve from a given state to any orthogonal state is h/(4\ensuremathΔE)h/(4\mathrm{\ensuremath{\Delta}}E)h/(4\ensuremathΔE), where \ensuremathΔE\mathrm{\ensuremath{\Delta}}E\ensuremathΔE is the energy uncertainty. A related bound called the Margolus-Levitin theorem states that T\ensuremath⊥\ensuremath≥h/(2\ensuremath⟨E\ensuremath⟩){T}_{\ensuremath{\perp}}\ensuremath{\ge}h/(2\ensuremath{\langle}E\ensuremath{\rangle})T\ensuremath⊥​\ensuremath≥h/(2\ensuremath⟨E\ensuremath⟩), where \ensuremath⟨E\ensuremath⟩\ensuremath{\langle}E\ensuremath{\rangle}\ensuremath⟨E\ensuremath⟩ is the expectation value of energy and the ground energy is taken to be zero. Many subsequent works have interpreted T\ensuremath⊥{T}_{\ensuremath{\perp}}T\ensuremath⊥​ as defining a minimal time for an elementary computational operation and correspondingly a fundamental limit on clock speed determined by a system's energy. Here we present local time-independent Hamiltonians in which computational clock speed becomes arbitrarily large relative to \ensuremath⟨E\ensuremath⟩\ensuremath{\langle}E\ensuremath{\rangle}\ensuremath⟨E\ensuremath⟩ and \ensuremathΔE\mathrm{\ensuremath{\Delta}}E\ensuremathΔE as the number of computational steps goes to infinity. We argue that energy considerations alone are not sufficient to obtain an upper bound on computational speed, and that additional physical assumptions such as limits to information density and information transmission speed are necessary to obtain such a bound.
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
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.
Cloud-based quantum computers do not provide users with access to hardware-level information such as the underlying Hamiltonians, which obstructs the characterization of their physical properties. We propose a method to infer the energy scales of gate Hamiltonians in such black-box quantum processors using only user-accessible data, by exploiting quantum speed limits. Specifically, we reinterpret the Margolus-Levitin and Mandelstam-Tamm bounds as estimators of the energy expectation value and variance, respectively, and relate them to the shortest time for the processor to orthogonalize a quantum state. This shortest gate time, expected to lie on the nanosecond scale, is inferred from job execution times measured in seconds by employing gate-time amplification. We apply the method to IBM's superconducting quantum processor and estimate the energy scales associated with single-, two-, and three-qubit gates. The order of estimated energy is consistent with typical drive energies in superconducting qubit systems, suggesting that current gate operations approach the quantum speed limit. Our results demonstrate that fundamental energetic properties of black-box quantum computers can be quantitatively accessed through operational time measurements, reflecting the conjugate relationship between time and energy imposed by the uncertainty principle.
Quantum speed limits set the maximal pace of state evolution. Two well-known limits exist for a unitary time-independent Hamiltonian: the Mandelstam-Tamm and Margolus-Levitin bounds. The former restricts the rate according to the state energy uncertainty, while the latter depends on the mean energy relative to the ground state. Here we report on an additional bound that exists for states with a bounded energy spectrum. This bound is dual to the Margolus-Levitin one in the sense that it depends on the difference between the state's mean energy and the energy of the highest occupied eigenstate. Each of the three bounds can become the most restrictive one, depending on the spread and mean of the energy, forming three dynamical regimes which are accessible in a multilevel system. The new bound is relevant for quantum information applications, since in most of them, information is stored and manipulated in a Hilbert space with a bounded energy spectrum.
How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide and can be attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit. These results establish the fundamental limit of the operation rate of any information processing system.