Quantum speed limit theorems

Quantum speed limit theorems are quantum mechanics theorems concerning the orthogonalization interval, the minimum time for a quantum system to evolve between two orthogonal states, also known as the quantum speed limit.

Consider an initial pure quantum state expressed as a superposition of its energy eigenstates

$\left|\psi (0)\right\rangle =\sum _{n}c_{n}\left|E_{n}\right\rangle$.

If the state $\left|\psi (0)\right\rangle$ is let to evolve for an interval $\delta t$ by the Schrödinger equation it becomes

$\left|\psi (\delta t)\right\rangle =\sum _{n}c_{n}e^{-i{\frac {E_{n}\delta t}{\hbar }}}\left|E_{n}\right\rangle$,

where $\hbar ={\frac {h}{2\pi }}$ is the reduced Planck constant. If the initial state $\left|\psi (0)\right\rangle$ is orthogonal to the evolved state $\left|\psi (\delta t)\right\rangle$ then $\left\langle \psi (0)|\psi (\delta t)\right\rangle =0$ and the minimum interval $\delta t_{\perp }$ required to achieve this condition is called the orthogonalization interval or time.

Mandelstam-Tamm theorem

The Mandelstam-Tamm theorem states that

$\delta E\delta t_{\perp }\geq \hbar {\frac {\pi }{2}}$,

where

$(\delta E)^{2}=\left\langle \psi |H^{2}|\psi \right\rangle -(\left\langle \psi |H|\psi \right\rangle )^{2}={\frac {1}{2}}\sum _{n,m}|c_{n}|^{2}|c_{m}|^{2}(E_{n}-E_{m})^{2}$,

is the variance of the system's energy and $H$ is the Hamiltonian operator. The theorem is named after Leonid Mandelstam and Igor Tamm. In this case, quantum evolution is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space; it is the distance along this curve measured by the Fubini-Study metric.

Proof

We want to find the smallest interval $\delta t_{\perp }$ such that

$|S(\delta t_{\perp })|^{2}=|\left\langle \psi (0)|\psi (\delta t_{\perp })\right\rangle |^{2}=0$.

We note that

${\begin{aligned}|S(\delta t)|^{2}&=|\left\langle \psi (0)|\psi (\delta t)\right\rangle |^{2}=\sum _{n,m}|c_{n}|^{2}|c_{m}|^{2}e^{-i{\frac {\delta t}{\hbar }}\left(E_{n}-E_{m}\right)}=\\&=\sum _{n,m}|c_{n}|^{2}|c_{m}|^{2}\cos \left({\frac {\delta t}{\hbar }}\left(E_{n}-E_{m}\right)\right),\end{aligned}}$

using Euler's formula and noting that the sine function is odd. Then

${\begin{aligned}|S(\delta t)|^{2}&\geq 1-{\frac {4}{\pi ^{2}}}\sum _{n,m}|c_{n}|^{2}|c_{m}|^{2}{\frac {\delta t}{\hbar }}\left(E_{n}-E_{m}\right)\sin \left({\frac {\delta t}{\hbar }}\left(E_{n}-E_{m}\right)\right)\\&-{\frac {2}{\pi ^{2}}}\sum _{n,m}|c_{n}|^{2}|c_{m}|^{2}\left({\frac {\delta t}{\hbar }}\left(E_{n}-E_{m}\right)\right)^{2}\end{aligned}}$,

since $\cos(x)\geq 1-{\frac {4}{\pi ^{2}}}x\sin(x)-{\frac {2}{\pi ^{2}}}x^{2}$, $\forall x\in \mathbb {R}$. We note that

${\frac {d|S(\delta t)|^{2}}{d\delta t}}=-\sum _{n,m}|c_{n}|^{2}|c_{m}|^{2}\sin \left({\frac {\delta t}{\hbar }}\left(E_{n}-E_{m}\right)\right){\frac {E_{n}-E_{m}}{\hbar }}$.

Thus

$|S(\delta t)|^{2}\geq 1+{\frac {4}{\pi ^{2}}}\delta t{\frac {d|S(\delta t)|^{2}}{d\delta t}}-{\frac {1}{\pi ^{2}}}\left({\frac {2\delta t}{\hbar }}\delta E\right)^{2}$.

Since $|S(\delta t)|^{2}\geq 0$ then ${\frac {d|S(\delta t)|^{2}}{d\delta t}}=0$ if $S(\delta t)=0$. So the second term vanishes for $\delta t=\delta t_{\perp }$ and

$0\geq 1-{\frac {1}{\pi ^{2}}}{\frac {4\delta t_{\perp }^{2}}{\hbar ^{2}}}\left(\delta E\right)^{2}$.

For this bound to become an equality we demand $\cos(x)=1-{\frac {4}{\pi ^{2}}}x\sin(x)-{\frac {2}{\pi ^{2}}}x^{2}$, that is $x=0$ or $x=\pm \pi$. Thus

${\frac {\delta t_{\perp }}{\hbar }}\left(E_{n}-E_{m}\right)=0\quad {\text{or}}\quad {\frac {\delta t_{\perp }}{\hbar }}\left(E_{n}-E_{m}\right)=\pm \pi \quad \forall n,m,c_{n}\neq 0,c_{m}\neq 0$,

which holds for only two energy eigenstates $E_{0}=0$ and $E_{1}=\pm {\frac {\pi \hbar }{\delta t_{\perp }}}$. Thus, the only state that attains this bound is a two-level pure quantum state (qubit) in an equal superposition

$\left|\psi _{q}\right\rangle ={\frac {1}{\sqrt {2}}}\left(e^{i\varphi _{0}}\left|0\right\rangle +e^{i\varphi _{1}}\left|\pm {\frac {\pi \hbar }{\delta t_{\perp }}}\right\rangle \right)$

of energy eigenstates $\left|E_{0}\right\rangle$ and $\left|E_{1}\right\rangle$, unique up to degeneracy of the energy level $E_{1}$ and arbitrary phase factors $\varphi _{0}$, $\varphi _{1}$ of the eigenstates.

Margolus–Levitin theorem

The Margolus–Levitin theorem states that

$E_{avg}\delta t_{\perp }\geq \hbar {\frac {\pi }{2}}$,

where

$E_{avg}=\left\langle \psi |H|\psi \right\rangle =\sum _{n}|c_{n}|^{2}E_{n}$,

is the system's average energy and $H$ is the Hamiltonian operator, such that

• $H$ does not depend on time;
• $H$ has zero ground state energy.

The theorem is named after Norman Margolus and Lev B. Levitin.

Proof

Graphs of trigonometric functions used in inequalities of Mandelstam-Tamm and Margolus–Levitin theorems.

We want to find the smallest interval $\delta t_{\perp }$ such that

$S(\delta t_{\perp })=\left\langle \psi (0)|\psi (\delta t_{\perp })\right\rangle =\sum _{n}|c_{n}|^{2}e^{-i{\frac {E_{n}\delta t_{\perp }}{\hbar }}}=0$.

We note that

${\begin{aligned}{\text{Re}}(S(\delta t))&=\sum _{n}|c_{n}|^{2}\cos \left({\frac {E_{n}\delta t}{\hbar }}\right)\geq \\&\geq \sum _{n}|c_{n}|^{2}\left(1-{\frac {2}{\pi }}{\frac {E_{n}\delta t}{\hbar }}-{\frac {2}{\pi }}\sin \left({\frac {E_{n}\delta t}{\hbar }}\right)\right)=\\&=\sum _{n}|c_{n}|^{2}-{\frac {2\delta t}{\pi \hbar }}\sum _{n}|c_{n}|^{2}E_{n}-{\frac {2}{\pi }}\sum _{n}|c_{n}|^{2}\sin \left({\frac {E_{n}\delta t}{\hbar }}\right)=\\&=1-{\frac {2\delta t}{\pi \hbar }}E_{avg}+{\frac {2}{\pi }}{\text{Im}}(S(\delta t))\end{aligned}}$,

as $\cos(x)\geq 1-{\frac {2}{\pi }}x-{\frac {2}{\pi }}\sin(x),\forall x\geq 0$. Since $S(\delta t_{\perp })=0$ requires ${\text{Re}}(S(\delta t_{\perp }))={\text{Im}}(S(\delta t_{\perp }))=0$ then

$0\geq 1-{\frac {2}{\pi }}{\frac {E_{avg}\delta t_{\perp }}{\hbar }}$.

For this bound to become an equality we demand $\cos(x)=1-{\frac {2}{\pi }}(x+\sin(x))$, that is $x=0$ or $x=\pi$. Thus

${\frac {E_{n}\delta t_{\perp }}{\hbar }}=0\quad {\text{or}}\quad {\frac {E_{n}\delta t_{\perp }}{\hbar }}=\pi \quad \forall n,c_{n}\neq 0$,

which holds for only two energy eigenstates $E_{0}=0$ and $E_{1}={\frac {\pi \hbar }{\delta t_{\perp }}}$. Thus, the only state that attains this bound is a two-level pure quantum state (qubit) in an equal superposition

$\left|\psi _{q}\right\rangle ={\frac {1}{\sqrt {2}}}\left(e^{i\varphi _{0}}\left|0\right\rangle +e^{i\varphi _{1}}\left|{\frac {\pi \hbar }{\delta t_{\perp }}}\right\rangle \right)$

of energy eigenstates $\left|E_{0}\right\rangle$ and $\left|E_{1}\right\rangle$, unique up to degeneracy of the energy level $E_{1}$ and arbitrary phase factors $\varphi _{0}$, $\varphi _{1}$ of the eigenstates.

Time-varying Hamiltonian

The Margolus-Levitin theorem generalizes to the case with time-varying Hamiltonian and mixed states.

Let $H_{\delta t}$ be the Hamiltonian at time interval $\delta t$, such that $H_{\delta t}$ still has zero energy in the ground state. Let the system start at some mixed state with density operator $\rho _{0}$ and evolve by the Schrödinger equation over time. Then

$\int _{0}^{\delta t}|tr(\rho _{0}H_{\delta t})|dt\geq \hbar D_{B}(\rho _{0},\rho _{\delta t})$,

where $D_{B}$ is the Bures distance between the starting state and the ending state.

To obtain the original theorem, set $H_{\delta t}$ to be independent of time, and $\rho _{0}=\left|\psi (0)\right\rangle \left\langle \psi (0)\right|$, then since pure states evolve to pure states, $\rho _{\delta t}=\left|\psi (\delta t)\right\rangle \left\langle \psi (\delta t)\right|$, and so by the formula for the Bures distance between pure states,

$E_{avg}\delta t\geq \hbar \arccos |\left\langle \psi (0)|\psi (\delta t)\right\rangle |$,

and when the starting and ending states are orthogonal, we obtain $E_{avg}\delta t_{\perp }\geq \hbar {\frac {\pi }{2}}$. However, the Margolus–Levitin theorem has not yet been established in time-dependent quantum systems, whose Hamiltonians $H_{\delta t}$ are driven by arbitrary time-dependent parameters, except for the adiabatic case.

Other relevant theorems

Relevant theorems concerning the Margolus–Levitin and the Mandelstam-Tamm theorems were proved in 2009 by Lev B. Levitin and Tommaso Toffoli.

Theorem

In the case $E_{avg}\neq \delta E$ the orthogonalization interval satisfies

$\delta t_{\perp }\leq {\frac {\pi \hbar \left(1+e^{\ln {\left|{\frac {\delta E}{E_{avg}}}\right|}}\right)}{2E_{avg}\left(1+{\frac {\delta E}{E_{avg}}}\right)}}\left(1+\epsilon \right)={\frac {\pi \hbar }{2E_{avg}}}\left(1+\epsilon \right),\quad \forall \epsilon >0$

Theorem

For any state $\left|\psi \right\rangle$

${\frac {E_{max}}{4}}\leq E_{avg}\leq {\frac {E_{max}}{2}}$,

where $E_{max}$ is the maximum energy eigenvalue of $\left|\psi \right\rangle$ and

$\pi \hbar \leq E_{max}\delta t_{\perp }\leq 2\pi \hbar$,

wherein $E_{max}\delta t_{\perp }=\pi \hbar$ for the qubit state $\left|\psi _{q}\right\rangle$ with $E_{1}=E_{max}$.

Proof

Let

$S(\delta t_{\perp })=\left\langle \psi (0)|\psi (\delta t_{\perp })\right\rangle =\sum _{n}|c_{n}|^{2}e^{-i{\frac {\delta t_{\perp }}{\hbar }}E_{n}}=0$.

Assume a contrario that $E_{max}>{\frac {2\pi \hbar }{\delta t_{\perp }}}$. We can define $E_{l}\doteq E_{max}-{\frac {2\pi \hbar }{\delta t_{\perp }}}>0$. But then

$e^{-i{\frac {\delta t_{\perp }}{\hbar }}E_{l}}=e^{-i{\frac {\delta t_{\perp }}{\hbar }}E_{max}}e^{2\pi i}=e^{-i{\frac {\delta t_{\perp }}{\hbar }}E_{max}}$.

Thus, replacing $E_{max}$ with $E_{l}>E_{max}$ does not change $S(\delta t_{\perp })$ and therefore the set of energy eigenvalues is bounded from above. To prove the existence of the lower bound on $E_{max}$, let the average energy be $E_{avg}^{(1)}$. We note that replacing energy levels $E_{n}$ in $S(\delta t_{\perp })$ with $E_{max}-E_{n}$ will not affect its validity. But after such a replacement, the average energy is $E_{avg}^{(2)}=E_{max}-E_{avg}^{(1)}$, and we can choose $E_{avg}=\min \left(E_{avg}^{(1)},E_{avg}^{(2)}\right)$. Thus $E_{avg}\leq {\frac {E_{max}}{2}}$. Using the bound on $E_{avg}$ from the Margolus–Levitin theorem completes the proof.

Furthermore, if $\delta t_{\perp }={\frac {\pi \hbar }{E_{max}}}$ then

$S(\delta t_{\perp })=\sum _{n=0}^{m}|c_{n}|^{2}e^{-i\pi {\frac {E_{n}}{E_{max}}}}=\sum _{n=0}^{m}|c_{n}|^{2}\left(\cos \left(\pi {\frac {E_{n}}{E_{max}}}\right)-i\sin \left(\pi {\frac {E_{n}}{E_{max}}}\right)\right)=0$,

which is satisfied iff $E_{0}=0$, $E_{1}=E_{max}={\frac {\pi \hbar }{\delta t_{\perp }}}$, and $|c_{n}|^{2}={\frac {1}{2}}$.

See also

• Bekenstein bound
• Bremermann's limit
• Landauer's principle
• Kolmogorov complexity
• Koomey's law
• Limits to computation
• Moore's law
• Uncertainty principle

References

• Deffner, Sebastian; Campbell, Steve (2017), "Quantum speed limits", Journal of Physics A, 50 (45): 453001, arXiv:1705.08023, Bibcode:2017JPhA...50S3001D, doi:10.1088/1751-8121/aa86c6, S2CID 3477317
• Jordan, Stephen P. (2017), "Fast quantum computation at arbitrarily low energy", Physical Review A, 95 (3): 032305, arXiv:1701.01175, Bibcode:2017PhRvA..95c2305J, doi:10.1103/PhysRevA.95.032305, S2CID 118953874
• Lloyd, Seth; Ng, Y. Jack, "Black Hole Computers", Scientific American (April 2007), p. 53–61
• Sinitsyn, Nikolai A. (2018). "Is there a quantum limit on speed of computation?". Physics Letters A. 382 (7): 477–481. arXiv:1701.05550. Bibcode:2018PhLA..382..477S. doi:10.1016/j.physleta.2017.12.042. S2CID 55887738.