Quantum speed limit

In quantum mechanics, a quantum speed limit (QSL) is a limitation on the minimum time for a quantum system to evolve between two distinguishable states.^[1] QSL are closely related to time-energy uncertainty relations. In 1945, Leonid Mandelstam and Igor Tamm derived a time-energy uncertainty relation that bounds the speed of evolution in terms of the energy dispersion.^[2] Over half a century later, Norman Margolus and Lev Levitin showed that the speed of evolution cannot exceed the mean energy,^[3] a result known as the Margolus–Levitin theorem. Realistic physical systems in contact with an environment are known as open quantum systems and their evolution is also subject to QSL.^[4][5] Quite remarkably it was shown that environmental effects, such as non-Markovian dynamics can speed up quantum processes,^[6] which was verified in a cavity QED experiment.^[7]

QSL have been used to explore the limits of computation^[8][9] and complexity. In 2017, QSLs were studied in a quantum oscillator at high temperature. ^[10] In 2018, it was shown that QSL are not restricted to the quantum domain and that similar bounds hold in classical systems. ^[11][12] In 2021, both the Mandelstam-Tamm and the Margolus-Levitin QSL bounds were concurrently tested in a single experiment^[13] which indicated there are "two different regimes: one where the Mandelstam-Tamm limit constrains the evolution at all times, and a second where a crossover to the Margolus-Levitin limit occurs at longer times."

Mandelstam-Tamm limit

Let [math]\displaystyle{ ds^2 }[/math] be the Bures metric, defined by[math]\displaystyle{ ds(\rho, \rho+d\rho)^2 = \frac 12 \sum_{jk} \frac{|\langle j|d\rho |k\rangle|^2}{p_j + p_k} }[/math]If a quantum system is evolving under a time-dependent Hamiltonian [math]\displaystyle{ H }[/math], then its velocity according to Bures metric is upper bounded by[math]\displaystyle{ ds \leq \frac{\sigma_H(t)}{\hbar}dt }[/math]where [math]\displaystyle{ \sigma_H(t) }[/math] is the uncertainty in energy at time [math]\displaystyle{ t }[/math].

Two corollaries:

• The time taken to evolve from [math]\displaystyle{ \rho }[/math] to [math]\displaystyle{ \rho' }[/math] is [math]\displaystyle{ \tau \geq \frac{\hbar}{\bar\sigma_H}dist(\rho, \rho') }[/math], where is [math]\displaystyle{ \bar \sigma_H := \frac{1}{\tau}\int_0^\tau \sigma_H(t)dt }[/math] the time-averaged uncertainty in energy.
• The time taken to evolve from one pure state to another pure state orthogonal to it is [math]\displaystyle{ \tau \geq \frac\pi 2 \frac{\hbar}{\bar\sigma_H} }[/math]. ^[14]

Applications

Computation machinery is constructed out of physical matter that follows quantum mechanics, and each operation, if it is to be unambiguous, must be a transition of the system from one state to an orthogonal state. Suppose the computing machinery is a physical system evolving under Hamiltonian that does not change with time. Then according to the Margolus–Levitin theorem, its operations per time per energy obey [math]\displaystyle{ \frac{1/\delta t_{\perp}}{E} \le \frac{1}{\hbar \frac\pi 2} = 6 \times 10^{33} \mathrm{s}^{-1}\cdot \mathrm{J}^{-1}. }[/math] That is, the processing rate of all forms of computation cannot be higher than about 6 × 10³³ operations per second per joule of energy. This includes "classical" computers since even classical computers are still made of matter that follows quantum mechanics.

References

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