# From quantum speed limits to energy-efficient quantum gates

Maxwell Aifer1,3 and Sebastian Deffner1,2 1Department of Physics, University
of Maryland, Baltimore County, Baltimore, MD 21250, USA  2Instituto de Física
‘Gleb Wataghin’, Universidade Estadual de Campinas, 13083-859, Campinas, São
Paulo, Brazil  3Author to whom any correspondence should be addressed.
<EMAIL_ADDRESS>
###### Abstract

While recent breakthroughs in quantum computing promise the nascence of the
quantum information age, quantum states remain delicate to control. Moreover,
the required energy budget for large scale quantum applications has only
sparely been considered. Addressing either of these issues necessitates a
careful study of the most energetically efficient implementation of elementary
quantum operations. In the present analysis, we show that this optimal control
problem can be solved within the powerful framework of quantum speed limits.
To this end, we derive state-independent lower bounds on the energetic cost,
from which we find the universally optimal implementation of unitary quantum
gates, for both single and $N$-qubit operations.

##  1 Introduction

The promise of exploiting quantum features in already existing or near-term
technologies [1, 2, 3, 4] for useful applications has led to increasing
funding of quantum initiatives and agendas [5, 6, 7, 8, 9], with global
investment in quantum computing research expected to grow by over 10% annually
through 2027 [10]. Remarkably, during the same period, global spending on
quantum computing products is predicted to increase by over 50% annually [10].
In anticipation of this growth, it seems important to clearly define the
“quantum promise”, both with regards to its impact on society [11, 12] and
with regards to its physical and technological ramifications.

Mathematically motivated progress towards an answer to this question has been
made in the field of computational complexity, where problems are classified
by the amount of resources necessary to solve them on classical and quantum
computers. It has been shown, for instance, that an integer can be factored
using a quantum computer in an amount of time which is upper bounded by a
polynomial function of the number of digits the integer has [13]. To the very
best of our knowledge, to date no algorithm has been published which runs on a
classical computer and solves the same problem in polynomial time. Another
example of a theoretical “quantum speedup” is a quantum algorithm for training
perceptrons, a fundamental task in machine learning, which requires an amount
of time scaling with the square root of the number of data points [14]. The
best known classical algorithm requires an amount of time which is linear in
the number of data points [14].

While the asymptotic scaling of resource usage with problem size on quantum
computers is increasingly well understood [15], there has been less emphasis
placed on calculating the coefficients which determine this dependence
exactly; that is, it may be known that an algorithm can be executed in an
amount of time which is proportional to a particular function of the input
size, but the constant of proportionality is seldom known precisely. This is
partly because the coefficients differ from one machine to another, and are
therefore not fundamentally properties of the algorithm [16]. An additional
wrinkle is that the execution time is often estimated by reference to the
“gate complexity” [17], which is the number of gates needed to implement an
algorithm in the quantum circuit model. Depending on which set of gates one
takes to be the smallest building blocks of the algorithm (called “elementary
operations”), the coefficients may vary.

The importance of these coefficients is underscored by current efforts to
create exascale classical computers around the world [18]. These systems will
make new computations feasible, not by decreasing the number of operations
necessary to complete the task, but by dramatically reducing the time cost per
operation. It is also informative to compare the current monetary cost of
elementary operations on classical and quantum computers. Recent benchmarking
of the Amazon Web Services (AWS) computing platform shows that the platform
charges about $4\times 10^{-13}$ cents per classical floating point operation,
whereas a single quantum circuit evaluation costs $1$ cent on an AWS-owned
Rigetti machine [19, 20]. Given such a pricing scheme, while it is certainly
true that there must be some problem size above which an $O(\sqrt{N})$ quantum
algorithm will be more cost-effective than an $O(N)$ classical algorithm, it
is far from obvious that this problem size is exceeded in practical use-cases.

It is therefore desirable to find a measure of the cost of running a quantum
algorithm which can be precisely quantified, independent of the specific
hardware used. As stated earlier, the execution time of an algorithm is
hardware-dependent, but this is not to say that the execution time can be
decreased at will without incurring some other cost [15]. In particular,
quantifying the amount of energy required to operate quantum technologies is
crucially important [15]. It is interesting to note that quantifying energy
usage for computing is an inherently important area of research. The issue of
energy-efficiency in computing has recently received increased attention due
to the high stakes involved for Earth’s climate. For instance, regulators have
announced that goals set in the Paris Climate Agreement are not likely to be
reached unless bitcoin mining is banned across all of Europe, due to the large
energy consumption of mining algorithms [21]. Another concerning revelation is
that “training a single AI model can emit as much carbon as five cars in their
lifetimes”[22].

Thus, as a first and foundational step towards addressing this important
problem, we pose the question: “What is the minimum amount of energy needed to
implement a quantum gate in a set amount of time?” We will answer this
question for general quantum gates of arbitrary dimension, and will provide an
explicit expression for a protocol which attains this minimum energetic cost
in the form of a Hamiltonian $H$. We find that in general the minimum
energetic cost is nonzero, and depends on the particular gate being
implemented. Moreover, this lower bound cannot be violated by any machine.
Since the elementary operations in the circuit model are individual gates, our
findings can be used to efficiently implement a quantum algorithm consisting
of many gates. Our analysis is rooted in an active field of modern research
that is focused on so-called quantum speed limits (QSLs) [23, 24, 25, 26].

These QSLs are a collection of theorems which originate in mathematically
rigorous treatments of Heisenberg’s uncertainty relation for energy and time
[27]. While originally only formulated for isolated, undriven dynamics [28,
29, 30, 31], in recent years QSLs have been generalized to controlled dynamics
[32, 33] and open quantum systems [34, 35, 36, 37, 38]. For a comprehensive
review we refer to the literature [23, 24, 25]. In principle, studying the QSL
has given rise to two fundamentally different questions [39], namely either to
quantify the minimal time a quantum system needs to evolve between distinct
states [40, 41, 42, 43, 44, 45], or to bound the maximal rate with which
quantum states can evolve [46, 47, 48, 49]. While either version of the QSL
can be exploited to assess the time cost and energy cost of a quantum
computation [50, 51, 52], previous formulations are limited by the fact that
the QSL has been considered as an inherent property of the evolving state –
and not of the Hamiltonian generating the dynamics.

Thus, our analysis starts with a derivation of a new QSL, which is independent
of the evolving state, and is fully determined by the eigenvalues of the
governing Hamiltonian. This is essential to quantify the cost of a quantum
gate, independent of the input and output states. To this end, several new
quantities need to be defined. Each quantum gate has an associated “shortest
covering arc length”, which is a nonnegative scalar that measures how
difficult it is to implement the gate. We prove a theorem which states that
the minimum energetic cost of implementing a gate is proportional to its
shortest covering arc length. Moreover, we also derive a lower bound on the
closely related “energy-time phase volume” of evolution. As a main result, we
formulate a “recipe” to design the most energetically efficient implementation
of single as well as $N$-qubit gates.

##  2 Speed Limits and Energetic Cost

We begin with a mini-review of notions and notations. This will motivate and
inform how to quantify the energetic cost of any (unitary) quantum operation.
Ideally, we would like a quantum computer to go between distinguishable states
as quickly as possible and while using as little energy as possible.
Mathematically, the question may be posed as follows: suppose that at time
$t=0$ the system is in a known state $\left|\psi_{0}\right\rangle$. The system
is assumed to be isolated from the rest of the universe, and has a constant
Hamiltonian $H$, and hence the state of the system evolves according to the
Schrödinger equation,
$i\hbar\partial_{t}\left|\psi\right\rangle=H\left|\psi\right\rangle$. Subject
to these assumptions, what is the smallest possible time $\tau$ such that
$\left\langle\psi(0)|\psi(\tau)\right\rangle=0$?

The minimal time to reach an orthogonal state is then given by the
Mandelstamm-Tamm (MT) inequality [28],$\tau\geq\hbar\pi/2\Delta E$, where
$\Delta E$ is the standard deviation in energy of the initial state. A
seemingly independent bound on the orthogonalization time is given by the
Margolus-Levitin (ML) inequality [29], $\tau\geq\hbar\pi/2\left(\langle
E\rangle-E_{0}\right)$, where $\langle
E\rangle=\left\langle\psi\right|H\left|\psi\right\rangle$ and $E_{0}$ is the
ground state energy. However, it has been noted that the “unified” bound is
tight [31], and hence the QSL time for orthogonal states can be written as

$\tau\geq\tau_{\mathrm{QSL}}\equiv\max\left\\{\frac{\hbar\pi/2}{\Delta
E},\frac{\hbar\pi/2}{\langle E\rangle-E_{0}}\right\\}.$ (1)

Many, but not all, computations have the property that they leave the computer
in a state which can be distinguished from its original state. Two pure states
$\left|\psi_{1}\right\rangle$ and $\left|\psi_{2}\right\rangle$ can be
perfectly distinguished from each other if and only if they are orthogonal
[53, 54]. However, in the study of QSLs it is often instructive to consider
arbitrary angles [55], and hence less then perfect distinguishability. The
angle between two pure states is simply defined as

$\mathcal{B}(\psi_{1},\psi_{2})=\arccos{\left|\left\langle\psi_{1}|\psi_{2}\right\rangle\right|}.$
(2)

Choosing some angle $\theta$, we then ask what is the smallest $\tau$ for
which $\mathcal{B}(\psi_{0},\psi(\tau))=\theta$. A tight bound is given by a
straightforward generalization of Eq. (1) [56],

$\tau\geq\tau_{\mathrm{QSL}}\equiv\max\left\\{\frac{\hbar\theta}{\Delta
E},\frac{\hbar\theta}{\langle E\rangle-E_{0}}\right\\}.$ (3)

However, computing the such defined $\tau_{\mathrm{QSL}}$ requires knowledge
of both the initial and final states.

More typically for a computer, the initial and final states are unknown. In
this case it is not possible to evaluate $\Delta E$ and $\langle E\rangle-
E_{0}$. However, these quantities can be upper bounded using the fact that the
energy eigenvalues are necessarily bounded for a finite-dimensional system.
Consider $N$ energy eigenvalues that are sorted in increasing order
($E_{0}\leq E_{1}\leq\dots E_{N-1}$), and we assume without loss of generality
that $E_{0}\geq 0$111If the ground state energy was negative, $E_{0}<0$, we
could always “shift” the energy spectrum by including a simple additive term
in the Hamiltonian. Now observe that since $\langle E\rangle$ is at most
$E_{N-1}$, the least upper bound on $\langle E\rangle-E_{0}$ is
$E_{N-1}-E_{0}$. This leads to the following “state-independent” ML inequality

$\tau\geq\frac{\hbar\theta}{E_{N-1}-E_{0}}.$ (4)

We proceed similarly with the MT result. A theorem known as Popoviciu’s
inequality [57] asserts that $\Delta E\leq(E_{N-1}-E_{0})/2$, and so the
corresponding state-independent MT inequality is

$\tau\geq\frac{2\hbar\theta}{E_{N-1}-E_{0}}.$ (5)

Equation (5) is our first main result. As desired, this speed limit can be
computed from the Hamiltonian alone, without needing to know the initial
state. Whereas Eq. (3) states that a _specific_ state cannot travel an angle
$\theta$ in time less than $\tau$, Eq. (5) says that _no_ state can travel an
angle $\theta$ in time less than $\tau$. Equation (4) is also a valid speed
limit, but since it is less restrictive than Eq. (5), it is not practically
useful. Equation (5), in turn, is less restrictive than Eq. (3); however, it
has the advantage that it can be computed when the system’s state is not
specified.

A generalization of (5) is obtained by allowing for a time-dependent
Hamiltonian $H(t)$. We assume that for some large integer $M$, if the interval
$(0,\tau)$ is subdivided into $M$ segments of equal length, then the
Hamiltonian is approximately constant within each segment. Applying Eq. (5) in
each segment gives

$\frac{\tau}{M}(E_{N-1}(t_{j})-E_{0}(t_{j}))\geq 2\hbar\,\delta\theta_{j},$
(6)

Where $t_{j}=j\tau/M$ and
$\delta\theta_{n}=\mathcal{B}(\psi(t_{n}),\psi(t_{n+1}))$ Summing over all
segments, we obtain

$\sum_{j=1}^{M}\frac{\tau}{M}(E_{N-1}(t_{j})-E_{0}(t_{j}))\geq
2\hbar\sum_{j=1}^{M}\delta\theta_{j}.$ (7)

Let $\theta$ again be the angle between the initial and final states, i.e.
$\theta=\mathcal{B}(\psi_{0},\psi(\tau))$. It is easy to see that
$\mathcal{B}(\cdot,\cdot)$ is a distance function which satisfies the triangle
inequality [54]. Therefore $\theta$ cannot exceed the sum of the angles
$\delta\theta_{n}$. Using this fact and taking the limit of $M\to\infty$ gives

$A\equiv\int_{0}^{\tau}dt\,(E_{N-1}(t)-E_{0}(t))\geq 2\hbar\theta.$ (8)

Equation (8) states that when the energy eigenvalues are plotted as functions
of time, the area contained between the graphs of the largest and smallest
eigenvalues must be at least $2\hbar\theta$, see Fig. 1 for an illustration.
It should be noted that in the single qubit case $2\theta$ is the angle
between the initial and final states on the Bloch sphere. Due to this
graphical interpretation, we refer to the quantity on the left hand side of
Eq. (8) as the energy-time phase volume, and denote it by $A$.

Figure 1:  The area contained in the shaded region is bounded below by the
distinguishability between the initial and final states, compare Eq. (8).

Equations (5) and (8) can be used to derive corresponding inequalities in
terms of the Schatten-$p$-norm of the Hamiltonian,

$\|H\|_{p}=\left(\sum_{n=0}^{N-1}|E_{n}|^{p}\right)^{1/p}.$ (9)

It is clear from the definition (9) that $\|H\|_{p}\geq|E_{n}|$ for all $n$,
which implies that $E_{N-1}-E_{0}\leq 2\|H\|_{p}$ for all $p$.

Figure 2: Comparison of three different estimates of energy spread for a state
with an arbitrarily-chosen energy distribution. In red we depict the energy
distribution, the yellow shaded area corresponds to the standard deviation,
and the grey shaded area is the “maximal” spread of energy eigenvalues. Note
that the “maximal spread” is clearly smaller than twice the operator norm
$\|H\|$.

In particular, taking $p$ to infinity gives the operator norm, defined as the
maximum of the absolute values of the energy eigenvalues

$\|H\|=\max\\{|E_{0}|,|E_{1}|,\dots,|E_{N-1}|\\},$ (10)

and for which we still have $E_{N-1}-E_{0}\leq 2\|H\|$, see Fig. 2 for an
illustration. Therefore, from (5), we immediately obtain the inequality

$\tau\geq\frac{\hbar\theta}{\|H\|_{p}},$ (11)

valid for all $p$, including $p=\infty$. Using Eq. (11) in the derivation of
Eq. (8) we obtain

$\int_{0}^{\tau}dt\,\|H(t)\|_{p}\geq\hbar\theta.$ (12)

The quantity appearing on the left hand side of Eq. (12) generalizes the
energetic cost functional first introduced in Ref. [58], and then used by Abah
et al. [59] to assess different methods of shortcuts to adiabaticity [60]. It
is interesting to note that this quantity does correspond in some cases to the
amount of energy required to control a quantum system. For instance, for a
qubit in a magnetic field, the energy used to create the field is proportional
to this expression. Therefore, we will henceforth refer to this quantity as
simply the “energetic cost”. Unless otherwise specified, we set $p=\infty$,
which provides a lower bound for all other cases. Interestingly, the same
energetic cost appeared in Ref. [61], in a quantum analog of Landauer’s
principle for unitary quantum evolution.

As was shown above, the energetic cost gives a general form of the quantum
speed limit. Having dimensions of energy times time, the energetic cost is
formally an action [62]. When it is minimized using variational methods, the
formalism becomes analogous to Lagrangian mechanics, a parallel which is
highlighted in Sec. 5.

As was explained above, Eq. (12) is useful when the initial state of the
system is ambiguous, as opposed to Eq. (1), which is useful when the initial
state is specified. An important application where the initial state is
generally unspecified is the operation of quantum gates. A quantum gate is a
unitary operator $U$ acting on the Hilbert space $\mathcal{H}$, which maps
each state $\left|\psi\right\rangle$ to a state $U\left|\psi\right\rangle$. We
would like to have a “gate quantum speed limit”, which describes the energy-
time trade-off necessary for operating a quantum gate. However, Eq. (12) is
still not suitable for this purpose, since the angle $\theta$ is also
undetermined in the case of quantum gates. This is due to the fact that the
distinguishability between the initial and final states depends on what the
initial state is. In Sec. 3, we will identify the correct quantity to take the
place of $\theta$ for single qubit gates, which will be generalized to gates
of arbitrary dimension in Sec. 4.

##  3 Energy-efficient single qubit gates

Our main objective is now to find the minimum energetic cost of implementing
such single qubit gates as well as the corresponding optimal protocol. To this
end, consider a unitary operator $U$ in the form of a $2\times 2$ matrix, and
we want to find a Hamiltonian $H(t)$ such that for some time $\tau$,
independent of the initial state $\left|\psi(0)\right\rangle$, we will have

$\left|\psi(\tau)\right\rangle=\exp{\left(i\varphi\right)}\,U\left|\psi(0)\right\rangle,$
(13)

for some undetermined real number $\varphi$. We allow for an arbitrary overall
phase factor $\exp{\left(i\varphi\right)}$, since we consider operations on a
single qubit in isolation. Although $\varphi$ may take any value, its value
must be the same regardless of the initial state. Constructing such
Hamiltonians can be done systematically by exploiting techniques from inverse
engineering [63]. For our purposes, we we will call a time-dependent
Hamiltonian an _allowed protocol_ , if the corresponding dynamics satisfy
(13). We are then interested in minimizing the energetic cost functional over
the set of all allowed protocols,

$C[H]=\int_{0}^{\tau}dt\,\|H(t)\|.$ (14)

Any allowed protocol which minimizes $C$ is called an _optimal protocol_. The
problem is then to find the minimum cost and at least one optimal protocol,
given an arbitrary unitary operator $U$.

###  3.1 Recasting the problem in terms of Bloch sphere rotations

For the single qubit case, the Hamiltonian $H(t)$ takes the form of a $2\times
2$ Hermitian matrix and it can be expanded in the basis which consists of the
three Pauli matrices $(\sigma_{x},\sigma_{y},\sigma_{z})$ along with the
identity matrix $\mathbb{I}$. We have

$H(t)=u_{0}(t)\mathbb{I}+u_{1}(t)\sigma_{x}+u_{2}(t)\sigma_{y}+u_{3}(t)\sigma_{z}\,.$
(15)

The instantaneous eigenvalues of this Hamiltonian are then

$E_{\pm}=u_{0}(t)\pm\sqrt{u_{1}(t)^{2}+u_{2}(t)^{2}+u_{3}(t)^{2}}\,.$ (16)

This implies that the operator norm of the Hamiltonian simply reads

$\|H(t)\|=|u_{0}(t)|+\sqrt{u_{1}(t)^{2}+u_{2}(t)^{2}+u_{3}(t)^{2}}.$ (17)

It can be shown that varying $u_{0}(t)$ only affects the overall phase of the
state as it evolves (see Appendix 7.1). Therefore for any allowed protocol
with nonzero $u_{0}(t)$, we can obtain a new allowed protocol which has a
lower cost by setting $u_{0}(t)=0$ everywhere.

Next, we observe that if the wavefunction at time $t$ is
$\left|\psi(t)\right\rangle$, then a short time later, at time $t+dt$, the
wavefunction will be

$\left|\psi(t+dt)\right\rangle=\left(1-\frac{i}{\hbar}\,H(t)dt\right)\left|\psi(t)\right\rangle=\left(1-\frac{i}{\hbar}\,\vec{\sigma}\cdot\vec{u}(t)\,dt\right)\left|\psi(t)\right\rangle$
(18)

Equation (18) is simply an expression of the evolution operator to first order
in time. Comparing this with the definition of the infinitesimal rotation
$R(\hat{n},d\phi)$ [64], we immediately recognize

$\left|\psi(t+dt)\right\rangle=\hat{R}\left(\hat{u}(t),\frac{2|\vec{u}(t)|}{\hbar}dt\right)\left|\psi(t)\right\rangle,$
(19)

which means that $\left|\psi(t+dt)\right\rangle$ evolves by rotating
$\left|\psi(t)\right\rangle$ by an infinitesimal angle
$2|\vec{u}(t)|/\hbar\,dt$ around the $\hat{u}(t)$-axis. As always,
$\hat{u}(t)\equiv\vec{u}(t)/|\vec{u}(t)|$. It is equivalent to say that the
Bloch vector rotates with angular velocity $\vec{\omega}\equiv
2\vec{u}/\hbar$. Thus, it follows from Eq. (17) that the operator norm of the
Hamiltonian is simply given by $\|H(t)\|=\hbar|\vec{\omega}(t)|/2$, and we can
therefore express the cost functional (14) as

$C=\frac{\hbar}{2}\int_{0}^{\tau}dt\left|\vec{\omega}(t)\right|.$ (20)

By Euler’s rotation theorem, a sequence of rotations is itself a rotation.
Therefore the transformation accomplished by any unitary evolution can be
described (up to global phase) by a single rotation of the Bloch sphere about
an axis $\hat{n}$ by an angle $\alpha$. To determine the parameters $\alpha$
and $\hat{n}$ we compare the form of the unitary operator $U$ with the
definition of the rotation operator. Since we are ignoring the overall phase,
we may take $U$ to have determinant $1$. In this case $U$ must take the form
[65]

$U=\begin{pmatrix}a&b\\\ -b^{*}&a^{*}\end{pmatrix},$ (21)

where $|a|^{2}+|b|^{2}=1$. A general rotation operator with axis $\hat{n}$ and
angle $\alpha$ is given by

$R=\begin{pmatrix}\cos(\alpha/2)-in_{3}\sin(\alpha/2)&\sin(\alpha/2)(-n_{2}-in_{1})\\\
\sin(\alpha/2)(n_{2}-in_{1})&\cos(\alpha/2)+in_{3}\sin(\alpha/2)\end{pmatrix}$
(22)

Identifying the elements of the matrices in Eqs. (21) and (22), we obtain
expressions for the parameters $n_{1}$, $n_{2}$, $n_{3}$, and $\alpha$ in
terms of the matrix elements $a$ and $b$. There is still some ambiguity in
choosing the parameters $\hat{n}$ and $\alpha$, since the rotations
$R(\hat{n},\alpha)$ and $R(\hat{n},\alpha\pm n2\pi)$ are equivalent for all
$n\in\mathbb{Z}$. Similarly, the rotations $R(\hat{n},\alpha)$ and
$R(-\hat{n},2\pi-\alpha)$ are equivalent. We eliminate this ambiguity by
requiring that $0\leq\alpha\leq\pi$. Subject to this condition, we have

$\alpha=2\arccos(\text{Re}[a])\quad\text{and}\quad\hat{n}=-\frac{1}{\sin(\alpha/2)}\begin{pmatrix}\text{Im}[b]\\\
\text{Re}[b]\\\ \text{Im}[a]\end{pmatrix}.$ (23)

In conclusion, we have translated finding energetically optimal quantum gates
to a simple geometric problem on the Bloch sphere.

###  3.2 Solution to the single qubit problem

To find the actual solution of the optimal control problem, we now need to
determine a function $\vec{\omega}(t)$ such that if each point on the Bloch
sphere moves with angular velocity $\vec{\omega}$ from time $t=0$ to time
$\tau$, then the end result will be a rotation of the Bloch sphere by an angle
$\alpha$ about an axis $\hat{n}$. Of all such functions, we must find one
which minimizes Eq. (20), and then compute the minimal value of the cost.

A natural guess, which turns out to be correct, is that the rotation can be
implemented with minimal cost by rotating about the axis $\hat{n}$ at a
constant rate222Note that we are searching for the optimal protocol that
drives the quantum state along a geodesic of the Riemannian manifold
corresponding to the cost functional (14) [66, 67, 68, 62].. We will now show
that the following protocol minimizes the cost,
$\vec{\omega}(t)=\alpha/\tau\,\hat{n}$. First, it is apparent that this is an
allowed protocol since the axis of rotation is $\hat{n}$ and the sphere
rotates by an angle $\alpha$ between time $0$ and time $\tau$, so this process
results in the correct transformation of each state. By plugging the constant
protocol into Eq. (20) we see that the cost for this protocol simply is
$C=\hbar\alpha/2$.

What remains is to show that no other protocol can have a lower cost. To this
end, consider a vector $\vec{v}_{0}$ which is perpendicular to $\hat{n}$. Let
$\vec{v}(t)$ be the trajectory of the point on the Bloch sphere which starts
at $\vec{v}_{0}$. After the Bloch sphere has been rotated, the angle between
$\vec{v}(\tau)$ and $\vec{v}_{0}$ will be $\alpha$.

This simple geometric picture then makes the argument almost trivial. The
geodesics on the surface of a sphere are the great circles. Thus, the length
of the path traveled by the vector during the interval $(0,\tau)$ is at least
$\alpha$. Note that this would not necessarily be true if $\alpha$ were
allowed to be greater than $\pi$. The fact that $\alpha$ is required to be
less than $\pi$ is equivalent to assuming that the Bloch sphere is rotated
“the short way around” in order to reach the desired orientation, see Fig. 3.

Figure 3: Two paths on the Bloch sphere that accomplish the same rotation, but
one is longer

The integral of the speed of the vector $\vec{v}$ on the surface on the Bloch
sphere is therefore at least $\alpha$, that is

$\int_{0}^{\tau}dt\left|\frac{d\vec{v}}{dt}\right|\geq\alpha.$ (24)

Now exploiting the vector identity
$\partial_{t}\vec{v}=\vec{\omega}\times\vec{v}$, we have

$\int_{0}^{\tau}dt\,\left|\vec{\omega}\times\vec{v}\right|\geq\alpha\,,\quad\text{and
thus}\quad\int_{0}^{\tau}dt|\vec{\omega}(t)|\geq\alpha,$ (25)

where we used that $|\vec{\omega}||\vec{v}|\geq|\vec{\omega}\times\vec{v}|$.
Therefore, we immediately obtain a lower bound on the energetic cost,

$\int_{0}^{\tau}dt\,\|H(t)\|\geq\hbar\alpha/2,$ (26)

which is exactly what we set out to prove.

Equation (26) can be thought of as a QSL for single qubit gates. This is
especially apparent when it is put in a form more similar to the ML and MT
inequalities,

$\tau\geq\frac{\hbar\alpha}{2\langle\|H\|\rangle_{t}},$ (27)

where $\langle\cdot\rangle_{t}$ denotes time averaging. It is useful to
compare this result to other QSL results to better understand how it fits into
this body of literature. The MT and ML bounds describe the minimum time
necessary to get a system to go between a pair of _known_ states. In Sec. 2 we
derived a state-independent QSL in Eqs. (5), (11), and (12). These results
give the minimum time needed to go between a pair of _unknown_ states which
are separated by a _known_ angle. Finally, Eq. (27) is a QSL for arbitrary
quantum gates. This bounds gives the minimum time necessary to _implement a
quantum gate_ on an _unknown_ state, such that the system will end up in a
state which is separated from the first by an angle which is also unknown.

Comparing Eq. (26) with Eq. (12), we observe that for a quantum gate the
parameter $\alpha/2$ plays the same role that the angle $\theta$ plays for the
transition between two pure states. This can be understood by considering that
the angle $\theta$ is equal to half the angle between states on the Bloch
sphere. When a sphere is rotated about some axis by an angle $\alpha$, the
maximum angle which a point on the sphere goes through is $\alpha$, which
implies that

$\frac{\alpha}{2}=\max_{\psi_{0}}\left\\{\mathcal{B}(\left|\psi_{0}\right\rangle,U\left|\psi_{0}\right\rangle)\right\\}.$
(28)

Hence, $\alpha$ can be understood as a measure of the difficulty of
implementing a given unitary gate, since it sets the minimally required
energetic cost.

###  3.3 Pedagogical example: the Hadamard gate

The Hadamard gate is a single qubit gate, which we write as

$G=\frac{1}{\sqrt{2}}\begin{pmatrix}-i&&-i\\\ -i&&i\\\ \end{pmatrix}.$ (29)

Note that we chose a representation for which $\det{(G)}=1$. It is then easy
to see that we have

$\alpha=2\arccos(0)=\pi\quad\text{and}\quad\hat{n}=-\frac{1}{\sin(\pi/2)}\begin{pmatrix}-1/\sqrt{2}\\\
0\\\ -1/\sqrt{2}\end{pmatrix}=\begin{pmatrix}1/\sqrt{2}\\\ 0\\\
1/\sqrt{2}\end{pmatrix}.$ (30)

Thus, the angular velocity simply becomes

$\vec{\omega}=\frac{\pi}{\sqrt{2}\tau}\begin{pmatrix}1\\\ 0\\\
1\end{pmatrix}.$ (31)

From the latter we can then construct the optimal Hamiltonian

$H=\frac{\pi\hbar}{2\sqrt{2}\tau}(\sigma_{x}+\sigma_{z})=\frac{\pi\hbar}{2\sqrt{2}\tau}\begin{pmatrix}0&1-i\\\
1+i&0\end{pmatrix}.$ (32)

It is interesting to note that this $H$ belongs to the class of Hamiltonians
found in Ref. [63], and is even identical to the “optimal” example studied in
Ref. [61]. However, whereas in Ref. [61] the optimization was done somewhat
heuristically, in the present analysis we have solved the optimal control
problem rigorously for any single qubit quantum gates.

##  4 Generalization to $N$-qubit gates

We now consider the case of a gate which operates on $N$-qubit states. Thus,
suppose that $G$ is a $2^{N}\times 2^{N}$ unitary matrix and let $U(t)$ be the
time-evolution operator, which satisfies the corresponding Schrödinger
equation, such that $U(0)=\mathbb{I}$ and $U(\tau)=G$. As before, we are
interested in finding an optimal implementation of $G$, which minimizes the
energetic cost functional (14).

In complete analogy to the single qubit case, we first need to derive a
version of the QSL that is governed by the eigenvalues of $U$. To this end, we
will show that the eigenvalues of $U$ vary at rates which are bounded above by
a function of the Hamiltonian, and therefore comparing the initial and final
eigenvalues of $U$ gives a lower bound on the time-integral of that function.

Infinitesimally, the time evolution is given by

$U(t+dt)=U(t)-\frac{i}{\hbar}\,H(t)U(t)\,dt\,,$ (33)

which can be solved by means of first-order perturbation theory [69]. Thus, we
write

$\lambda_{n}(t+dt)=\lambda_{n}(t)-\frac{i}{\hbar}\left\langle\phi_{n}(t)\right|H(t)U(t)\left|\phi_{n}(t)\right\rangle
dt,$ (34)

where $\lambda_{n}(t)$ and $\left|\phi_{n}(t)\right\rangle$ are the
instantaneous eigenvalues and eigenvectors of $U(t)$. Therefore, we also have

$\frac{\lambda_{n}(t+dt)-\lambda_{n}(t)}{dt}=-\frac{i}{\hbar}\left\langle\phi_{n}(t)\right|H(t)\left|\phi_{n}(t)\right\rangle\lambda_{n}(t),$
(35)

Since $U(t)$ is unitary, its eigenvalues all have unit modulus, and we can
write $\lambda_{n}(t)=\exp{\left(i\theta_{n}(t)\right)}$. Inserting this into
Eq. (35) and taking the limit as $dt\to 0$ gives

$\frac{d\theta_{n}}{dt}=-\frac{1}{\hbar}\left\langle\phi_{n}(t)\right|H(t)\left|\phi_{n}(t)\right\rangle.$
(36)

The form of Eq. (36) is appealing, although it suffers from the issue that the
derivative of $\theta_{n}$ is undefined when $\theta_{n}$ crosses $2\pi$ and
jumps back to zero. To avoid this issue, we define a closely related333The
quantity $\omega_{n}$ can be interpreted as the speed at which the eigenvalue
$\lambda_{n}$ travels along the edge of the unit circle. It differs from
$d\theta_{n}/dt$ only because the derivative of $\theta_{n}$ is sometimes
undefined. quantity $\omega_{n}$ as

$\omega_{n}=-\frac{i}{\lambda_{n}(t)}\frac{d\lambda_{n}}{dt}.$ (37)

It is easy to see that $\omega_{n}$ has the beneficial qualities that it is
always defined and it is equal to the time-derivative of $\theta_{n}$ whenever
the latter exists. From Eq. (35) we then obtain

$\omega_{n}=-\frac{1}{\hbar}\left\langle\phi_{n}(t)\right|H(t)\left|\phi_{n}(t)\right\rangle.$
(38)

From the above it is apparent that for all $n$,

$-E_{N-1}\leq\hbar\omega_{n}\leq-E_{0}.$ (39)

It will prove convenient to now define a superoperator $L$ called the
“shortest covering arc length”. Given a unitary operator $X$, $L[X]$ is
defined to be the length of the shortest arc on the unit circle which includes
all the eigenvalues of $X$, see Fig. 4. It is clear that for a finite
dimensional operator $U$, the shortest covering arc must terminate on two of
its eigenvalues $\lambda_{\ell}$ and $\lambda_{k}$ for some $\ell$ and $k$.
Therefore

$\left|\frac{d}{dt}L[U(t)]\right|=|\omega_{k}(t)-\omega_{\ell}(t)|\,.$ (40)

Figure 4: The eigenvalues of the evolution operator must spread out by an
amount $L[G]$ between time $t=0$ and time $t=\tau$.

From Eq. (39) we then observe that

$\left|\frac{d}{dt}L[U(t)]\right|\leq\frac{E_{N-1}(t)}{\hbar}-\frac{E_{0}(t)}{\hbar}.$
(41)

Recall that $U(\tau)=G$ and $U(0)=\mathbb{I}$. It is evident that
$L[\mathbb{I}]=0$ since all of the eigenvalues of the identity coincide at
$1$. Finally, using the fundamental theorem of calculus we obtain

$\int_{0}^{\tau}dt\left(E_{N-1}(t)-E_{0}(t)\right)\geq\hbar L[G].$ (42)

This takes a particularly simple expression when put in terms of the energy-
time phase volume defined earlier, and we can write $A\geq\hbar L[G]$. Note
that

$\frac{1}{2}(E_{N-1}-E_{0})\leq\|H\|\leq E_{N-1}-E_{0}\,,$ (43)

and hence $A/2\leq C\leq A$. In conclusion, we have

$C=\int_{0}^{\tau}dt\,\|H(t)\|\geq\hbar L[G]/2$ (44)

Equation (44) is the minimum energetic cost required to implement a quantum
gate $G$, and thus it is a direct generalization of Eq. (26) to $N$-qubit
gates.

###  4.1 Optimal protocol

In complete analogy to before, we now proceed to find the optimal
implementation of the gate $G$, which attains the lower bound in Eq. (42). We
begin by assuming that for any gate $G$, there is some constant protocol which
is optimal, as was the case in the single qubit problem. That is, there is a
constant Hamiltonian $H$ such that $\exp{\left(-iH\tau/\hbar\right)}=G$, for
which we have

$\int_{0}^{\tau}dt\,(E_{N-1}(t)-E_{0}(t))=\hbar L[G]\,.$ (45)

Formally, $H$ can always be found by taking the matrix logarithm of $G$,
$H=i\hbar\ln(G)/\tau$. As before, all the angles $\theta_{i}$ are chosen to be
between $-\pi$ and $\pi$. Also note that since $G$ may be multiplied by an
arbitrary phase shift, we can assume that the largest and smallest angles are
chosen such that

$\theta_{N-1}-\theta_{0}=L[G],$ (46)

and such that $\theta_{0}=-\theta_{N-1}$. The eigenvalues of $H$ are then
simply $E_{n}=\hbar\theta_{n}/\tau$. It is easy to see that the energy-time
phase volume reads $A=\hbar(\theta_{N-1}-\theta_{0})$, and that the
corresponding energetic cost becomes $C=\hbar L[G]/2$. Thus, as was the case
for single qubit gates, optimal implementations of arbitrary $N$-qubit gates
are given by constant Hamiltonians.

###  4.2 Nonuniqueness of the optimal protocol

The immediate question arises, whether the constant protocol is the unique
optimal implementation, or whether there exist other protocols with the same,
minimal energetic cost. We will show now that there is is actually an infinite
family of optimal protocols, see also Refs. [70, 71, 61] for similar
conclusions.

Consider a function $f(t)$, which is everywhere nonnegative and satisfies
$\int_{0}^{\tau}dtf(t)=\tau$. Then let the Hamiltonian be given as a function
of time by $H^{\prime}(t)=f(t)i\hbar\ln(G)/\tau$ where the logarithm is chosen
as above. Then the evolution operator reads

$U(\tau)=\exp\left(\ln(G)\,\frac{1}{\tau}\int_{0}^{\tau}dt\,f(t)\right)=G,$
(47)

One easily convinces oneself that $H^{\prime}(t)$ is an allowed
implementation.

To see that this protocol is also optimal, note that since
$H^{\prime}(t)=f(t)H$ and $f(t)$ is nonnegative, we have
$\|H^{\prime}(t)\|=f(t)\|H\|$ and, hence,
$\int_{0}^{\tau}dt\|H^{\prime}(t)\|=\|H\|\int_{0}^{\tau}dtf(t)$. Thus, we
immediately have

$\int_{0}^{\tau}dt\,\|H^{\prime}(t)\|=\|H\|\int_{0}^{\tau}dt\,f(t)=\int_{0}^{\tau}dt\,\|H\|,$
(48)

which shows that $H^{\prime}(t)$ is optimal.

##  5 Higher order cost functionals

Having found an infinite family of implementations that are optimal with
respect to the energetic cost based on the operator norm (14) raises the
question whether working with other norms would given more insight or
restrictions. Thus, we conclude the analysis with generalizations of our
results to the Lebesgue $p$-norm and the Schatten $p$-norm for arbitrary $p$.

###  5.1 Lebesgue norm cost functional

We start with the Lebesgue $p$-norm, and consider the modified cost functional

$C_{p}[H]=\left(\int_{0}^{\tau}dt\,\|H(t)\|^{p}\right)^{1/p}.$ (49)

Given $p\geq 1$, Hölder’s inequality states that

$\int_{0}^{\tau}dt\,|f(t)g(t)|\leq\left(\int_{0}^{\tau}dt\,|f(t)|^{p}\right)^{1/p}\left(\int_{0}^{\tau}dt|g(t)|^{q}\right)^{1/q},$
(50)

where $q=1/(1-p^{-1})$. Setting $g(t)=1$, $f(t)=\|H(t)\|$ and $p=1$ we obtain

$\int_{0}^{\tau}dt\,\|H(t)\|\leq\left(\int_{0}^{\tau}dt\|H(t)\|^{p}\right)^{1/p}\tau^{1/q},$
(51)

which leads to a lower bound on $C_{p}$,

$C_{p}[H]\geq\tau^{-1/q}\,C_{1}[H].$ (52)

Then using the definition of $q$ and the lower bound on $C_{1}\geq\hbar
L[G]/2$ obtained above in Eq. (44), we can write

$C_{p}[H]\geq\frac{\hbar L[G]}{2}\,\tau^{(1-p)/p}.$ (53)

Hölder’s inequality becomes an equality iff the functions $f$ and $g$ are
linearly dependent. This means that the minimal cost is attained only if
$\|H(t)\|$ is constant. Thus, working with the Lebesgue $p$-norm identifies
the constant protocol as the _unique_ optimal implementation. It is
interesting to note that Eq. (53) gives a lower bound on the cost which
depends on protocol duration, unlike Eq. (44) which did not depend on the
protocol duration.

An interesting case is $p=2$ for a single qubit, for which we can write the
cost functional as

$C_{2}[H]=\sqrt{\int_{0}^{\tau}dt\,|\vec{\omega}(t)|^{2}}\,.$ (54)

Equation (54) is proportional to the square root of the dynamical action for a
rigid rotating sphere with no external potential. This implies that minimizing
the cost $C_{2}$ is equivalent to solving for the evolution of a rigid
rotating sphere between time $t=0$ and time $t=\tau$, subject to the
constraint that the initial and final orientations are specified. Thus, it is
no surprise that a constant-axis rotation with constant angular velocity gives
an optimal solution.

###  5.2 Schatten norm cost functional

Finally, we consider a generalized cost which is given by

$C_{p}[H]=\int_{0}^{\tau}dt\,\|H\|_{p},$ (55)

where $\|H\|_{p}$ is the Schatten $p$-norm (9). In this case, it will prove
instructive to demand that the gate $G$ is implemented exactly, without
allowing an arbitrary phase factor. That is, we require that $U(\tau)=G$. As
before, we parameterize the eigenvalues of $G$ as
$\lambda_{j}=\exp{\left(i\theta_{j}\right)}$, with $\theta_{j}$ between $-\pi$
and $\pi$. Now using the triangle inequality for the vector $p$-norm gives

$\int_{0}^{\tau}dt\,|\vec{\omega}(t)|_{p}\geq|\vec{\theta}(\tau)|_{p},$ (56)

where $|\vec{\theta}|_{p}=(\sum_{j}|\theta_{j}|^{p})^{1/p}$ and
$|\vec{\omega}|_{p}=(\sum_{j}|\omega_{j}|^{p})^{1/p}$. Using Eq. (38) we then
obtain

$|\vec{\omega}|_{p}=\frac{1}{\hbar}\left(\sum_{j}|\left\langle\phi_{j}|H|\phi_{j}\right\rangle|^{p}\right)^{1/p}\,,$
(57)

and since
$|\left\langle\phi_{j}|H|\phi_{j}\right\rangle|^{p}\leq\left\langle\phi_{j}||H|^{p}|\phi_{j}\right\rangle$,
we have

$|\vec{\omega}|_{p}\leq\frac{1}{\hbar}\left(\sum_{j}\left\langle\phi_{j}||H|^{p}|\phi_{j}\right\rangle\right)^{1/p}=\frac{1}{\hbar}\,\tr{(|H|^{p})^{1/p}}.$
(58)

The right hand side of Eq. (58) is simply an equivalent expression for the
Schatten $p$-norm. Hence, $|\vec{\omega}|_{p}\leq\|H\|_{p}/\hbar$, and we
obtain a lower bound on the cost

$\int_{0}^{\tau}dt\,\|H(t)\|_{p}\geq\hbar\,|\vec{\theta}|_{p}.$ (59)

Solving the optimal control problem for Eq. (59) is significantly more
involved, which is why we leave this issue for future work.

##  6 Concluding Remarks

The results in this paper build on recent progress in the field of QSL to give
physical constraints on the best-case performance of a quantum computer.
Specifically we have found the maximal energy efficiency of quantum gates, and
the protocol which attains that maximum. While energy-efficiency is not as
commonly discussed as certain other notions of performance (e.g. time-
complexity of algorithms), we believe that the energy efficiency is an
important figure of merit which deserves further study.

Historically, research on maximal energy efficiency of physical systems has a
rich tradition, perhaps the most celebrated example being Carnot’s
foundational contribution to thermodynamics. Since Carnot not only found the
maximal efficiency of a heat engine, but also provided a protocol which
attains this efficiency, his work served as a template for both physicists and
engineers. Ultimately this work made it possible to refine the creations of
the early industrial era into the sophisticated machines which are available
today.

A full thermodynamic understanding of quantum computing may prove similarly
revolutionary. In order to achieve such an understanding, it will be necessary
to combine bounds on best-case performance (such as QSLs) with a quantum
treatment of work and heat. For this reason, one of the more pressing
questions related to this work is: “How exactly is the energetic cost
functional related to the heat and work inputs and outputs of a quantum
system?” In addition to this question, a number of other questions remain to
be answered. An important consideration which we have neglected in this paper
is the cost of separating the system from the environment. Similarly, the use
of error-correcting protocols likely increases the minimal cost needed to
implement a quantum gate. These additional costs may represent a significant
portion of the actual energy needed to operate a quantum computer in practice,
which we leave to be investigated in future work.

## Acknowledgments

M.A. gratefully acknowledges support from Harry Shaw of NASA Goddard Space
Flight Center and Kenneth Cohen of Peraton.

## Appendix

###  7.1 Proof that identity term only contributes an overall phase

Recall that the differential equation for the evolution operator is

$U(t+dt)=\left(\mathbb{I}-\frac{i}{\hbar}H(t)dt\right)U(t)$ (A.1)

Putting in the Hamiltonian $H=u_{0}(t)\mathbb{I}+\vec{u}(t)\cdot\vec{\sigma}$
gives

$U(t+dt)=\left[\mathbb{I}-\frac{i}{\hbar}(u_{0}(t)\mathbb{I}+\vec{u}(t)\cdot\vec{\sigma})dt\right]U(t)dt.$
(A.2)

Separating the time interval into $M$ discrete steps, we then write the final
evolution operator $U(\tau)$ as a product

$U(\tau)=\prod_{j=1}^{M}\left[\mathbb{I}-\frac{i}{\hbar}(u_{0}(t_{j})\mathbb{I}+\vec{u}(t_{j})\cdot\vec{\sigma})dt\right].$
(A.3)

Note that to first order in $dt$,

$\left(\mathbb{I}-\frac{i}{\hbar}u_{0}\mathbb{I}dt\right)\left(\mathbb{I}-\frac{i}{\hbar}\vec{u}\cdot\vec{\sigma}dt\right)=\mathbb{I}-\frac{i}{\hbar}(u_{0}\mathbb{I}+\vec{u}\cdot\vec{\sigma})dt.$
(A.4)

Therefore we can factor each term in the product in this way, to obtain

$U(\tau)=\prod_{j=1}^{M}\left(\mathbb{I}-\frac{i}{\hbar}u_{0}(t_{j})\mathbb{I}dt\right)\left(\mathbb{I}-\frac{i}{\hbar}\vec{u}(t_{j})\cdot\vec{\sigma}dt\right).$
(A.5)

Since the first factor is a constant multiplied by the identity, it commutes
with any operator. Therefore

$U(\tau)=\prod_{j=1}^{M}\left(\mathbb{I}-\frac{i}{\hbar}\vec{u}(t_{j})\cdot\vec{\sigma}dt\right)\prod_{j=1}^{M}\left(\mathbb{I}-\frac{i}{\hbar}u_{0}(t_{j})\mathbb{I}dt\right).$
(A.6)

We then recognize the limit of the second product as a scalar exponential
multiplied by the identity operator

$\prod_{j=1}^{M}\left(\mathbb{I}-\frac{i}{\hbar}u_{0}(t_{j})\mathbb{I}dt\right)=\mathbb{I}\exp\left(-\frac{i}{\hbar}\int_{0}^{\tau}dtu_{0}(t)\right).$
(A.7)

Since the other factors in the evolution operator do not depend on $u_{0}$, we
have shown that $u_{0}$ only contributes an overall phase shift.

###  7.2 Proof that variable-axis protocols are non-optimal

It was shown in Sec. 3 that there is an optimal protocol for a single qubit
gate which rotates the Bloch sphere about a fixed axis. We now show that
protocols which rotate the Bloch sphere about an axis which is not fixed are
not optimal. Suppose that $\vec{v}_{0}$ is a vector perpendicular to
$\hat{n}$, and that $\vec{v}(t)$ is the trajectory of the point which is
initially at $\vec{v}_{0}$. Let $\vec{\omega}_{\parallel}(t)$ be the component
of $\vec{\omega}$ which is parallel $\vec{v}(t)$ and $\vec{\omega}_{\perp}(t)$
be the component of $\vec{\omega}$ which is perpendicular to $\vec{v}(t)$.
Note that

$\left|\frac{d\vec{v}}{dt}\right|=|\vec{\omega}_{\perp}|,$ (A.8)

so if $\omega_{\parallel}$ is not zero almost everywhere then

$\int_{0}^{\tau}dt|\vec{\omega}|=\int_{0}^{\tau}dt\sqrt{|\vec{\omega}_{\parallel}|^{2}+|\vec{\omega}_{\perp}|^{2}}>\int_{0}^{\tau}dt\sqrt{|\vec{\omega}_{\perp}|^{2}}=\int_{0}^{\tau}dt\left|\frac{d\vec{v}}{dt}\right|$
(A.9)

However it was already shown in Sec. 3 that

$\int_{0}^{\tau}dt\left|\frac{d\vec{v}}{dt}\right|\geq\alpha,$ (A.10)

which implies

$\int_{0}^{\tau}dt\,|\vec{\omega}|dt>\int_{0}^{\tau}dt\left|\frac{d\vec{v}}{dt}\right|\geq\alpha\,.$
(A.11)

Therefore this protocol cannot be optimal. Because the above argument applies
for any choice of $\vec{v}_{0}$ which is perpendicular to $\hat{n}$, the
protocol cannot be optimal unless $\vec{\omega}$ is always parallel to
$\hat{n}$, so the axis of rotation must be fixed.

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