Physics:Quantum speed limit

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In quantum mechanics, a quantum speed limit (QSL) is a limitation on the minimum time for a quantum system to evolve between two distinguishable (orthogonal) states. QSL theorems 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. Over half a century later, Norman Margolus and Lev Levitin showed that the speed of evolution cannot exceed the mean energy, 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. Quite remarkably it was shown that environmental effects, such as non-Markovian dynamics can speed up quantum processes, which was verified in a cavity QED experiment.

QSL have been used to explore the limits of computation and complexity. In 2017, QSLs were studied in a quantum oscillator at high temperature. In 2018, it was shown that QSL are not restricted to the quantum domain and that similar bounds hold in classical systems. In 2021, both the Mandelstam-Tamm and the Margolus–Levitin QSL bounds were concurrently tested in a single experiment 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."

In quantum sensing, QSLs impose fundamental constraints on the maximum achievable time resolution of quantum sensors. These limits stem from the requirement that quantum states must evolve to orthogonal states to extract precise information. For example, in applications like Ramsey interferometry, the QSL determines the minimum time required for phase accumulation during control sequences, directly impacting the sensor's temporal resolution and sensitivity.

A quantum speed limit sets a minimum time for a quantum state to evolve into a distinguishable state, linking the pace of quantum change to energy uncertainty and available energy.

Preliminary definitions

The speed limit theorems can be stated for pure states, and for mixed states; they take a simpler form for pure states. An arbitrary pure state can be written as a linear combination of energy eigenstates:

$|\psi ⟩=\sum _n{c}_n|E_n⟩.$

The task is to provide a lower bound for the time interval $t_{\perp }$ required for the initial state $|\psi ⟩$ to evolve into a state orthogonal to $|\psi ⟩$. The time evolution of a pure state is given by the Schrödinger equation:

$|{\psi }_t⟩=\sum _n{c}_n{e}^{it{E}_n/\hslash }|E_n⟩.$

Orthogonality is obtained when

$⟨{\psi }_0|{\psi }_t⟩=0$

and the minimum time interval $t=t_{\perp }$ required to achieve this condition is called the orthogonalization interval or orthogonalization time.

Mandelstam–Tamm limit

For pure states, the Mandelstam–Tamm theorem states that the minimum time $t_{\perp }$ required for a state to evolve into an orthogonal state is bounded below:

$t_{\perp }\ge \frac{\pi \hslash }{2\delta E}=\frac{h}{4\delta E}$,

where

$(\delta E{)}^2=⟨\psi |H^2|\psi ⟩-(⟨\psi |H|\psi ⟩{)}^2=\frac{1}{2}\sum _{n,m}|c_n{|}^2|c_m{|}^2(E_n-E_m{)}^2$,

is the variance of the system's energy and $H$ is the Hamiltonian operator. The quantum evolution is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space; the distance along this curve is measured by the Fubini–Study metric. This is sometimes called the quantum angle, as it can be understood as the arccos of the inner product of the initial and final states.

For mixed states

The Mandelstam–Tamm limit can also be stated for mixed states and for time-varying Hamiltonians. In this case, the Bures metric must be employed in place of the Fubini–Study metric. A mixed state can be understood as a sum over pure states, weighted by classical probabilities; likewise, the Bures metric is a weighted sum of the Fubini–Study metric. For a time-varying Hamiltonian $H_t$ and time-varying density matrix ${\rho }_t,$ the variance of the energy is given by

${\sigma }_H^2(t)=|\text{tr}({\rho }_t{H}_t^2)|-|\text{tr}({\rho }_t{H}_t){|}^2$

The Mandelstam–Tamm limit then takes the form

$\underset{0}{\overset{\tau }{\int }}{\sigma }_H(t)dt\ge \hslash D_B({\rho }_0,{\rho }_{\tau })$,

where $D_B$ is the Bures distance between the starting and ending states. The Bures distance is geodesic, giving the shortest possible distance of any continuous curve connecting two points, with ${\sigma }_H(t)$ understood as an infinitessimal path length along a curve parametrized by $t.$ Equivalently, the time $\tau$ taken to evolve from $\rho$ to ${\rho }^{\prime }$ is bounded as

$\tau \ge \frac{\hslash }{{\bar{\sigma }}_H}{D}_B(\rho ,{\rho }^{\prime })$

where

${\bar{\sigma }}_H=\frac{1}{\tau }\underset{0}{\overset{\tau }{\int }}{\sigma }_H(t)dt$

is the time-averaged uncertainty in energy. For a pure state evolving under a time-varying Hamiltonian, the time $\tau$ taken to evolve from one pure state to another pure state orthogonal to it is bounded as

$\tau \ge \frac{\hslash }{{\bar{\sigma }}_H}\frac{\pi }{2}$

This follows, as for a pure state, one has the density matrix ${\rho }_t=|{\psi }_t⟩⟨{\psi }_t|.$ The quantum angle (Fubini–Study distance) is then $D_B({\rho }_0,{\rho }_t)=\arccos |⟨{\psi }_0|{\psi }_t⟩|$ and so one concludes $D_B=\arccos 0=\pi /2$ when the initial and final states are orthogonal.

Margolus–Levitin limit

For the case of a pure state, Margolus and Levitin obtain a different limit, that

${\tau }_{\perp }\ge \frac{h}{4⟨E⟩},$

where $⟨E⟩$ is the average energy, $$⟨E⟩=E_{\text{avg}}=⟨\psi |H|\psi ⟩=\sum _n|c_n{|}^2{E}_n.$$ This form applies when the Hamiltonian is not time-dependent, and the ground-state energy is defined to be zero.

For time-varying states

The Margolus–Levitin theorem can also be generalized to the case where the Hamiltonian varies with time, and the system is described by a mixed state. In this form, it is given by

$\underset{0}{\overset{\tau }{\int }}|\text{tr}({\rho }_t{H}_t)|dt\ge \hslash D_B({\rho }_0,{\rho }_{\tau })$

with the ground-state defined so that it has energy zero at all times.

This provides a result for time varying states. Although it also provides a bound for mixed states, the bound (for mixed states) can be so loose as to be uninformative. The Margolus–Levitin theorem has not yet been experimentally established in time-dependent quantum systems, whose Hamiltonians $H_t$ are driven by arbitrary time-dependent parameters, except for the adiabatic case.

Dual Margolus–Levitin limit

In addition to the original Margolus–Levitin limit, a dual bound exists for quantum systems with a bounded energy spectrum. This dual bound, also known as the Ness–Alberti–Sagi limit or the Ness limit, depends on the difference between the state's mean energy and the energy of the highest occupied eigenstate. In bounded systems, the minimum time ${\tau }_{\perp }$ required for a state to evolve to an orthogonal state is bounded by

${\tau }_{\perp }\ge \frac{h}{4(E_{\max }-⟨E⟩)},$

where $E_{\max }$ is the energy of the highest occupied eigenstate and $⟨E⟩$ is the mean energy of the state. This bound complements the original Margolus–Levitin limit and the Mandelstam–Tamm limit, forming a trio of constraints on quantum evolution speed.

Levitin–Toffoli limit

A 2009 result by Lev B. Levitin and Tommaso Toffoli states that the precise bound for the Mandelstam–Tamm theorem is attained only for a qubit state. This is a two-level state in an equal superposition

$|{\psi }_q⟩=\frac{1}{\sqrt{2}}(|E_0⟩+e^{i\phi }|E_1⟩)$

for energy eigenstates $E_0=0$ and $E_1=\pm \pi \hslash /\Delta t$. The states $|E_0⟩$ and $|E_1⟩$ are unique up to degeneracy of the energy level $E_1$ and an arbitrary phase factor $\phi .$ This result is sharp, in that this state also satisfies the Margolus–Levitin bound, in that $E_{\text{avg}}=\delta E$ and so $t_{\perp }=\hslash \pi /2E_{\text{avg}}=\hslash \pi /2\delta E.$ This result establishes that the combined limits are strict:

$t_{\perp }\ge \max (\frac{\pi \hslash }{2\delta E},\frac{\pi \hslash }{2E_{\text{avg}}})$

Levitin and Toffoli also provide a bound for the average energy in terms of the maximum. For any pure state $|\psi ⟩,$ the average energy is bounded as

$\frac{E_{\text{max}}}{4}\le E_{\text{avg}}\le \frac{E_{\text{max}}}{2}$

where $E_{\text{max}}$ is the maximum energy eigenvalue appearing in $|\psi ⟩.$ (This is the quarter-pinched sphere theorem in disguise, transported to complex projective space.) Thus, one has the bound

$\frac{\pi \hslash }{E_{\text{max}}}\le t_{\perp }\le \frac{2\pi \hslash }{E_{\text{max}}}$

The strict lower bound $E_{\text{max}}{t}_{\perp }=\pi \hslash$ is again attained for the qubit state $|{\psi }_q⟩$ with $E_{\text{max}}=E_1$.

Bremermann's limit

Related topic: Bremermann's limit

The quantum speed limit bounds establish an upper bound at which computation can be performed. Computational 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, the number of operations per unit time per unit energy is bounded above by

$\frac{2}{\hslash \pi }=6\times 10^{33}{\mathrm{s}}^{-1}\cdot {\mathrm{J}}^{-1}$

This establishes a strict upper limit on the number of calculations that can be performed by physical matter. The processing rate of all forms of computation cannot be higher than about 6 × 10³³ operations per second per joule of energy. This is including "classical" computers, since even classical computers are still made of matter that follows quantum mechanics.

This bound is not merely a fanciful limit: it has practical ramifications for quantum-resistant cryptography. Imagining a computer operating at this limit, a brute-force search to break a 128-bit encryption key requires only modest resources. Brute-forcing a 256-bit key requires planetary-scale computers, while a brute-force search of 512-bit keys is effectively unattainable within the lifetime of the universe, even if galactic-sized computers were applied to the problem.

The Bekenstein bound limits the amount of information that can be stored within a volume of space. The maximal rate of change of information within that volume of space is given by the quantum speed limit. This product of limits is sometimes called the Bremermann–Bekenstein limit; it is saturated by Hawking radiation. That is, Hawking radiation is emitted at the maximal allowed rate set by these bounds.

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

9. Physics:Quantum mechanics measurements

10. Physics:Quantum state

11. Physics:Quantum system

12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

19. Physics:Quantum Measurement problem

20. Physics:Quantum Bell's theorem

21. Physics:Quantum Hidden variable theory

22. Physics:Quantum nonlocality

23. Physics:Quantum contextuality

24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

37. Physics:Quantum Runge–Lenz vector

38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (21) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Aharonov-Casher effect

77. Physics:Quantum Scattering theory

78. Physics:Quantum Scattering matrix

79. Physics:Quantum S-matrix

80. Physics:Quantum tunnelling

81. Physics:Quantum speed limit

82. Physics:Quantum revival

83. Physics:Quantum reflection

84. Physics:Quantum oscillations

85. Physics:Quantum jump

86. Physics:Quantum boomerang effect

87. Physics:Quantum chaos

7. Measurement and information (9) Back to index

88. Physics:Quantum Measurement theory

89. Physics:Quantum Measurement operators

90. Physics:Quantum Projective measurement

91. Physics:Quantum POVM

92. Physics:Quantum Weak measurement

93. Physics:Quantum Measurement collapse

94. Physics:Quantum entanglement

95. Physics:Quantum Zeno effect

96. Physics:Quantum limit

8. Quantum information and computing (15) Back to index

97. Physics:Quantum information theory

98. Physics:Quantum Qubit

99. Physics:Quantum Entanglement

100. Physics:Quantum Bell state

101. Physics:Quantum Gates and circuits

102. Physics:Quantum BB84

103. Physics:Quantum No-cloning theorem

104. Physics:Quantum Computing Algorithms in the NISQ Era

105. Physics:Quantum Noisy Qubits

106. Physics:Quantum error correction

107. Physics:Quantum Boson sampling

108. Physics:Quantum random access code

109. Physics:Quantum pseudo-telepathy

110. Physics:Quantum network

111. Physics:Quantum money

9. Quantum optics and experiments (10) Back to index

112. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

113. Physics:Quantum Stern-Gerlach experiment

114. Physics:Quantum Experimental quantum physics

115. Physics:Quantum optics

116. Physics:Quantum optics beam splitter experiments

117. Physics:Quantum Mach-Zehnder interferometer

118. Physics:Quantum Hong-Ou-Mandel effect

119. Physics:Quantum eraser experiment

120. Physics:Quantum delayed-choice quantum eraser

121. Physics:Quantum Ultra fast lasers

122. Template:Quantum optics operators

10. Open quantum systems (15) Back to index

123. Physics:Quantum Open systems

124. Physics:Quantum channel

125. Physics:Quantum Kraus operators

126. Physics:Quantum Amplitude damping

127. Physics:Quantum Phase damping

128. Physics:Quantum Depolarizing channel

129. Physics:Quantum Master equation

130. Physics:Quantum Lindblad equation

131. Physics:Quantum Decoherence

132. Physics:Quantum Dynamical decoupling

133. Physics:Quantum dissipation

134. Physics:Quantum Markov semigroup

135. Physics:Quantum Markovian dynamics

136. Physics:Quantum Non-Markovian dynamics

137. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

138. Physics:Quantum field theory (QFT) basics

139. Physics:Quantum field theory (QFT) core

140. Physics:Quantum Fields and Particles

141. Physics:Quantum Second quantization

142. Physics:Quantum Fock space

143. Physics:Quantum Harmonic Oscillator field modes

144. Physics:Quantum Creation and annihilation operators

145. Physics:Quantum vacuum fluctuations

146. Physics:Quantum Casimir effect

147. Physics:Quantum Propagators in quantum field theory

148. Physics:Quantum Feynman diagrams

149. Physics:Quantum Path integral formulation

150. Physics:Quantum Renormalization in field theory

151. Physics:Quantum Renormalization group

152. Physics:Quantum Field Theory Gauge symmetry

153. Physics:Quantum Spontaneous symmetry breaking

154. Physics:Quantum Non-Abelian gauge theory

155. Physics:Quantum Electrodynamics (QED)

156. Physics:Quantum chromodynamics (QCD)

157. Physics:Quantum Electroweak theory

158. Physics:Quantum Standard Model

159. Physics:Quantum triviality

160. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

161. Physics:Quantum Statistical mechanics

162. Physics:Quantum Partition function

163. Physics:Quantum Distribution functions

164. Physics:Quantum Liouville equation

165. Physics:Quantum Kinetic theory

166. Physics:Quantum Boltzmann equation

167. Physics:Quantum BBGKY hierarchy

168. Physics:Quantum Relaxation and thermalization

169. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (17) Back to index

170. Physics:Quantum Band structure

171. Physics:Quantum Fermi surfaces

172. Physics:Quantum Landau levels

173. Physics:Quantum fractional Hall effect

174. Physics:Quantum Semiconductor physics

175. Physics:Quantum Phonons

176. Physics:Quantum Electron-phonon interaction

177. Physics:Quantum Exchange interaction

178. Physics:Quantum Superconductivity

179. Physics:Quantum Topological phases of matter

180. Physics:Quantum anyon

181. Physics:Quantum well

182. Physics:Quantum spin liquid

183. Physics:Quantum spin Hall effect

184. Physics:Quantum phase transition

185. Physics:Quantum critical point

186. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

187. Physics:Quantum Fusion reactions and Lawson criterion

188. Physics:Quantum Plasma (fusion context)

189. Physics:Quantum Magnetic confinement fusion

190. Physics:Quantum Inertial confinement fusion

191. Physics:Quantum Plasma instabilities and turbulence

192. Physics:Quantum Tokamak core plasma

193. Physics:Quantum Tokamak edge physics and recycling asymmetries

194. Physics:Quantum Stellarator

15. Timeline (8) Back to index

195. Physics:Quantum mechanics/Timeline

196. Physics:Quantum mechanics/Timeline/Pre-quantum era

197. Physics:Quantum mechanics/Timeline/Old quantum theory

198. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

199. Physics:Quantum mechanics/Timeline/Quantum field theory era

200. Physics:Quantum mechanics/Timeline/Quantum information era

201. Physics:Quantum mechanics/Timeline/Quantum technology era

202. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

203. Physics:Quantum topology

204. Physics:Quantum battery

205. Physics:Quantum Supersymmetry

206. Physics:Quantum Black hole thermodynamics

207. Physics:Quantum Holographic principle

208. Physics:Quantum gravity

209. Physics:Quantum De Sitter invariant special relativity

210. Physics:Quantum Doubly special relativity

211. Physics:Quantum arithmetic geometry

212. Physics:Quantum unsolved problems

213. Physics:Quantum Yang-Mills mass gap

214. Physics:Quantum gravity problem

215. Physics:Quantum black hole information paradox

216. Physics:Quantum dark matter problem

217. Physics:Quantum neutrino mass problem

218. Physics:Quantum matter-antimatter asymmetry problem

Further reading

• Jordan, Stephen P. (2017). "Fast quantum computation at arbitrarily low energy". Physical Review A 95 (3). doi:10.1103/PhysRevA.95.032305. Bibcode: 2017PhRvA..95c2305J. 
• Sinitsyn, Nikolai A. (2018). "Is there a quantum limit on speed of computation?". Physics Letters A 382 (7): 477–481. doi:10.1016/j.physleta.2017.12.042. Bibcode: 2018PhLA..382..477S. 

Author: Harold Foppele

Source attribution: Physics:Quantum speed limit