Physics as Computing

About This Presentation

Title:

Physics as Computing

Description:

Physics as Computing Dr. Michael P. Frank Dept. of Electrical & Computer Eng. FAMU-FSU College of Engineering Quantum Computation for Physical Modeling Workshop – PowerPoint PPT presentation

Number of Views:102

Avg rating:3.0/5.0

Slides: 27

Provided by: Micha818

Learn more at: https://eng.fsu.edu

Category:

more less

Transcript and Presenter's Notes

Title: Physics as Computing

1
Physics as Computing

Dr. Michael P. FrankDept. of Electrical
Computer Eng.FAMU-FSU College of Engineering

Quantum Computation for Physical Modeling
Workshop(QCPM 04)Marthas VineyardWednesday,
September 15, 2004
2
Abstract

Studying the physical limits of computing
encourages us to think about physics in
computational terms.
Viewing physics as a computer directly gives us
limits on the computing capabilities of any
machine thats embedded within our physical
world.
We want to understand what various physical
quantities mean in a computational context.
Some answers so far
Entropy Unknown/incompressible information
Action Amount of computational work
Energy Rate of computing activity
Generalized temperature Clock frequency
(activity per bit)
Momentum Motional computation per unit
distance

Todaystopic
3
Entropy as Information

A bit of history
Most of the credit for originating this concept
really should go to Ludwig Boltzmann.
He (not Shannon) first characterized the entropy
of a system as the expected log-improbability of
its state, H -?(pi log pi).
He also discussed combinatorial reasons for its
increase in his famous H-theorem
Shannon brought Boltzmanns entropy to the
attention of communication engineers
And he taught us how to interpret Boltzmanns
entropy as unknown information, in a
communication-theory context.
von Neumann generalized Boltzmann entropy to
quantum mixed states
That is, the S -Tr ? ln ? expression that we
all know and love
Jaynes clarified how the von Neumann entropy of a
system can increase over time
Either when the Hamiltonian itself is unknown, or
when we trace out entangled subsystems
Zurek suggested adding algorithmically
incompressible information to the part of
physical information that we consider to be
entropy
I will discuss a variation on this theme.

4
Why go beyond the statistical definition of
entropy?

We may argue the statistical concept of entropy
is incomplete,
because it doesnt even begin to break down the
ontology-epistemology barrier
In the statistical view, a knower (such as
ourselves) must always be invoked to supply a
state of knowledge (probability distribution)
But we typically treat the knower as being
fundamentally separate from the physical system
itself.
However, in reality, we ourselves are part of the
physical system that is our universe
Thus, a complete understanding of entropy must
also address what knowledge means, physically

5
Small Physical Knowers

Of course, humans are extremely large complex
physical systems, and to physically characterize
our states of knowledge is a very long way off
However, we can hope to characterize the
knowledge of simpler systems.
Computer engineers find that in practice, it can
be very meaningful and useful to ascribe
epistemological states even to extremely simple
systems.
E.g., digital systems and their component
subsystems.
When analyzing digital systems,
we constantly say things like, At such-and-such
time, component A knows such-and-such information
about the state of component B
Means essentially that there is a specific
correlation between the states of A and B.
For nano-scale digital devices, we can strive to
exactly characterize their logical states in
mathematical physics terms..
Thus we ought to be able to say exactly what it
means for one component to know some information
about another.

6
What wed like to say

Want to formalize arguments such as the
following
Component A doesnt know the state of component
B, so the physical information in B is entropy to
component A. Component A cant destroy the
entropy in B, due to the 2nd law of
thermodynamics, and therefore A cant reset B to
a standard state without expelling Bs entropy to
the environment.
I want all of these to be mathematically
well-defined and physically meaningful
statements, and I want the argument to be
formally provable!
One motivation A lot of head-in-the-sand
technologists are still in a state of denial
about Landauers principle!
Oblivious erasure of non-entropy information
turns it into entropy.
We need to be able to prove it to them with
simple, undeniable, clear and correct arguments!
To get reversible/quantum computing more traction
in industry.

7
Insufficiency of Statistical Entropy for Physical
Knowers

Unfortunately for this kind of program
If the ordinary statistical definition of entropy
is used,
together with a knower that is fully defined as
an actual physical system, then
The 2nd law of thermodynamics no longer holds!
Note the unknown information in a system can be
reduced
Simply let the knower system perform a
(coherent, reversible) measurement of the target
system, to gain knowledge about the state of the
target system!
The entropy of the target system (from knowers
perspective) is then reduced.
The 2nd law says there must be a corresponding
increase in entropy somewhere, but where?
This is the essence of Maxwells Demon paradox.

8
Entropy in knowledge?

Resolution suggested by Bennett
The demons knowledge of the result of his
measurement can itself be considered to
constitute one form of entropy!
It must be expelled into environment in order to
reset his state.
But, what if we imagine ourselves in the demons
shoes?
Clearly, the demons knowledge of the measurement
result itself constitutes known information,
from his own perspective!
I.e., the demons own subjective posterior
probability distribution that he would (or
should) assess over the possible values of his
knowledge of the result, after he has already
obtained this knowledge, will be entirely
concentrated on the actual outcome.
The statistical entropy of this distribution is
zero!
So, here we have a type of entropy that is
present in our own knowledge itself, and is not
unknown information!
Needed A way to make sense of this, and to
mathematically quantify this entropy of
knowledge.

9
Quantifying the Entropy ofKnowledge, Approach 1

The traditional position says In order to
properly define the entropy in the demons state
of knowledge, we must always pop up to the
meta-perspective from which we are describing the
whole physical situation.
We ourselves always implicitly possess some
probability distribution over the states of the
joint demon-target system.
We should just take the statistical entropy of
that distribution.
Problem This approach doesnt face up to the
fact that we are physical systems too!
It doesnt offer any self-consistent way that
physical systems themselves can ever play the
role of a knower!
I.e., describe other systems, assess subjective
probability distributions over their state,
modify those distributions via measurements, etc.
This contradicts our own personal physical
experience,
as well as what we expect that quantum computers
performing coherent measurements of other systems
ought to be able to do

10
Approach 2

The entropy inherent in some known information is
the smallest size to which this information can
be compressed.
But of course, this depends on the coding system.
Zurek suggests, use Kolmogorov complexity. (Size
of shortest generating program.)
But there are two problems with doing that
Its only well-defined up to an additive
constant.
That is, modulo a choice of universal programming
language.
Its uncomputable!
What else might we try?

11
Approach 3 (We Suggest)

We propose The entropy content of some known
piece of information is its compressed size
according to whatever encoding would yield the
smallest expected compressed size, a priori.
That is, taking the expectation over all the
possible patterns of information before the
actual one was obtained.
This is nice, because the expected value of
posterior entropy then closely matches the
ordinary statistical entropy of the prior
distribution.
Even exactly, in special cases, or in the limit
of many repetitions
Due to a simple application of Shannons
channel-capacity theorem.
We can then show that the 2nd law gets obeyed on
average.
But, from whose a priori probability distribution
is this expectation value of compressed size to
be obtained?

Expected length of thecodeword ci
encodinginformation pattern i
12
Who picks the compressor?

Two possible answers to this
Use our probability distribution when we
originally describe and analyze the hypothetical
situation from outside.
Although this is a bit distasteful, since here we
are resorting to the meta-perspective again,
which we were trying to avoid
However, at least we do manage to sidestep the
paradox
Or, we can use the demons own a priori
assessment of the probabilities
That is, essentially, let him pick his own
compression system, however he wants!
The entropy of knowledge is then defined in a
relative way, as the smallest size that a given
entity with that knowledge would or could
compress that knowledge to,
given a specification of its capabilities,
together with any of its previous decisions
commitments as to the compression strategy it
would use.

13
A Simple Example

Suppose we have a seperable two-qubit system ab,
Where qubit a initially contains 1 bit of
entropy
I.e., described by density operator ?a ?0 ?1
0??0 1??1.
while qubit b is in a pure state (say 0?)
Its density operator (if we care) is ?b ?0
0??0.
Now, suppose we do a CNOT(a,b).
Can view this process as a measurement of qubit
a by qubit b.
Qubit b could be considered a subsystem of some
quantum know
Assuming the observer knows that this process has
occurred,
We can say that he now knows the state of a!
Since the state of a is now correlated with a
part of bs own state.
I.e., from bs personal subjective point of
view,bit a is no longer an unknown bit
But it is still entropy, because theexpected
compressed size of anencoding of this data is
still 1 bit!
This becomes clearer in a larger example

?a ?0?1
?ab ?00 ?01 00??00 11??11
?b 0?
14
Slightly Larger Example

Suppose system A initially contains 8 random
qubits a0a7, with a uniform distribution over
their values
a thus contains 8 bits of entropy.
And system B initially contains a large number
b0, of empty qubits.
b contains 0 entropy initially
Now, say we do CNOT(ai, bi) for i0 to 3
B now knows the values of a0,,a3.
The information in A that is unknown by b is now
only the 4 other bits a4a7.
But, the AB system also contains an additional 4
bits of information about A (shared between A and
B) which (though known by B) is (we expect) still
incompressible by B
I.e., the encoding that offers the minimum
expected length (prior to learning a0a3) still
has an expected length of 4 bits!
A second CNOT(bi, ai) can allow B to reversibly
clear the entropyfrom system A.
Note this is a Maxwells Demon type of scenario.
Entropy isnt lost because the incompressible
information in B is still entropy!
From an outside observers perspective, the
amount of unknown information remains the same in
all these situations
But from an inside perspective, entropy can flow
(reversibly) from known to unknown and back

15
Entropy Conversion
4 bits of Aknown to B(correlation)
Target system A
4 bits un-known to B
8 bits unknown to B
CNOT(a0-3?b0-3)
a0 a1 a2 a3 a4 a5 a6 a7
x0 x1 x2 x3 x4 x5 x6 x7
a0 a1 a2 a3 a4 a5 a6 a7
x0 x1 x2 x3 x4 x5 x6 x7
A
A
b0 b1 b2 b3 b4 b5 b6 b7
x0 x1 x2 x3 0 0 0 0
b0 b1 b2 b3 b4 b5 b6 b7
0 0 0 0 0 0 0 0
B (reversibly)measures A
B
B
Demon system B
4 bits of knowledge 8 bits all
together compressibleto 4 bits

In all stages, there remain 8 total bits of
entropy.
All 8 are unknown to us in our
meta-perspective.
But some may be known to subsystem B!
Still call them entropy for B if we dont
expect B can compress them away

4 bits un-known to B
a0 a1 a2 a3 a4 a5 a6 a7
0 0 0 0 x4 x5 x6 x7
A
CNOT(b0-3?a0-3)B (reversibly)controls A
b0 b1 b2 b3 b4 b5 b6 b7
x0 x1 x2 x3 0 0 0 0
B
4 incompressiblebits in Bs internalstate of
knowledge
16
Are we done?

I.e., have we arrived at a satisfactory
generalization of the entropy concept?
Perhaps not quite, because
Weve been vague about how to define the
compression system that the knower would use.
Or in other words, the knowers prior
distribution.
We havent yet provided an operational definition
(that can be replicably verified by a third
party) of the meaning of
The entropy of a physical system A, as assessed
by another physical system (the knower) B.
However, there might be no way to do better

17
One Possible Conclusion

Perhaps the entropy of a particular piece of
known information can only be defined relative to
a given description system.
Where by description system I mean a bijection
between compressed decompressed
informational objects ci ? di
Most usefully, the map should be computable.
This is not really any worse than the situation
with standard statistical entropy, where it is
only defined relative to a given state of
knowledge, in the form of a probability
distribution over states of the system.
The existence of optimal compression systems for
given probability distributions strengthens the
connection.
In fact, we can also infer a probability
distribution from the description system, in
cases of optimal description systems
We could consider a description system, rather
than a probability distribution, to be the
fundamental starting point for any discussion of
entropy.
Perhaps we should not be disappointed

18
The Entropy Game

A thought experiment to illustrate why
Suppose C wants to know Bs entropy for another
system A.
Classical protocol
B performs any desired

19
The Entropy Game

A thought experiment that can be used to
operationally define the entropy content of a
given target physical system X.
X should have a well-defined state space,
with N states total information content Itot
log N.
Basic idea B must use A (reversibly) as a
storage medium for data provided by C.
The entropy of C is defined as its total
info. content, minus the expected logarithm of
the number of messages that A can reliably store
and retrieve from it.

Rules of the game
A and B start out unentangled with each other
(and with C).
A publishes his own exact initial classical
state A0 in a public record.
B can probe A to make sure he is telling the
truth.
Meanwhile, B prepares in secret any string WW0
of any number n of bits.
B passes his string W to A. A may observe its
length n.

A may then carry out any fixed quantum algorithm
Q1 operating on the closed joint system (A,X,W),
under the condition
The final state must leave (A,X,W) unentangled,
AA0, and W 0n.
B is allowed to probe A and W to verify that
AA0 and W0n.
Finally, A carries out another fixed quantum
algorithm Q2, returning again to his initial
state A0, and supposedly restoring W to its
initial state.
A returns W to B B is allowed to check W and A
again to verify that these conditions are
satisfied.

Iterate till convergence.
Definition The entropy of system X is C minus
the maximum over As strategies (starting states
A0, and algorithms Q1,Q2) of the expectation
value (over states of X) of the minimum over Bs
strategies (sequences of strings) of the average
length of those strings that are exactly
returned by A (in step 8) with zero probability
of error.
20
The Entropy Game

A game (or adversarial protocol) between two
players (A and B) that can be used to
operationally define the entropy content of a
given target physical system X.
X should have a well-defined state space,
with N states total information content Itot
log N.
Basic idea B must use A (reversibly) as a
storage medium for data provided by C.
The entropy of C is defined as its total
info. content, minus the expected logarithm of
the number of messages that A can reliably store
and retrieve from it.

Rules of the game
A and B start out unentangled with each other
(and with C).
A publishes his own exact initial classical
state A0 in a public record.
B can probe A to make sure he is telling the
truth.
Meanwhile, B prepares in secret any string WW0
of any number n of bits.
B passes his string W to A. A may observe its
length n.

A may then carry out any fixed quantum algorithm
Q1 operating on the closed joint system (A,X,W),
under the condition
The final state must leave (A,X,W) unentangled,
AA0, and W 0n.
B is allowed to probe A and W to verify that
AA0 and W0n.
Finally, A carries out another fixed quantum
algorithm Q2, returning again to his initial
state A0, and supposedly restoring W to its
initial state.
A returns W to B B is allowed to check W and A
again to verify that these conditions are
satisfied.

A wants to show that X has a low entropy (high
available storage capacity or extropy).
He will choose an encoding of strings W in Xs
state that is as efficient as possible.
A chooses his strategy without knowledge of what
strings B will provide
The coding scheme must thus be very general.
Meanwhile, B wants to show that X has a high
entropy (low capacity).
B will

22
Explaining Entropy Increase

When the Hamiltonian of a closed system is
exactly known,
The statistical (von Neumann) entropy of the
systems density operator is exactly conserved.
I.e., there is no entropy increase.
In the traditional statistical view of entropy,
Entropy can only increase in one of the following
situations
(a) The Hamiltonian is not precisely known, or
(b) The system is not closed
Entropy can leak into the system from an outside
environment
(c) We estimate entropy by tracing over entangled
subsystems
Take reduced density operators of individual
subsystems
And pretend the entropy is additive
However, in the

23
Energy as Computing

Some history of the idea
Earliest hints can be seen in the original Planck
Eh? relation for light.
That is, an oscillation with a period of ?
requires an energy at least h?.
Also suggestive is the energy-time uncertainty
principle ?E?t ?/2.
Relates average energy uncertainty ?E to minimum
time intervals ?t.
Margolus Levitin, Physica D 120188-195 (1998).
Prove that a state of average energy E above the
ground state takes at least time ?t h/4E to
evolve to an orthogonal one.
Or (N-1)h/2NE, for a cycle of N mutually
orthogonal states.
Lloyd, Nature 4061047-1054, 31 Aug. 2000.
Uses that to calculate the maximum performance of
a 1 kg ultimate laptop.
Levitin, Toffoli, Walton, quant-ph/0210076.
Investigate minimum time to perform a CNOT
phase rotation, given E.
Giovannetti, Lloyd, Maccone, Phys. Rev. A 67,
052109 (2003), quant-ph/0210197 also see
quant-ph/0303085.
Tighter limits on time to reduce fidelity to a
given level, taking into account both E and ?E,
amount of entanglement, and number of interaction
terms.
These kinds of results prompt us to ask
Is there some valid sense in which we can say
that energy is computing?
And if so, what is it, exactly?
Well see this also relates to action as
computation.

24
A Simple Example

Consider a constant Hamiltonian with energy
eigenstates G? and E?, with eigenvalues 0,E.
That is, HG?0, HE?EE?. E.g., H ?Osz.
Consider the initial state ?0?
(G?E?)2-1/2.
cE? phase-rotates at rate ?E? E / ?.
In time 2E/h, rotates by ?p.
The area swept out by cE?(t) is
aE? ½p(cE?2) p / 4.
This is just ½ of a circle withradius rE?
2-1/2.
Meanwhile, cG? is stationary.
Sweeps out zero area.
Total area a p / 4.

i
ap/4
?p
1
cE?
cG?
0
r 2-1/2
25
Lets Look at Another Basis

Define a new basis 0?, 1? with
0?(G?E?)2-1/2, 1?(G?-E?)2-1/2
Use the same initial state ?0? 0?.
Note the final state is 1?.
Coefficients c0?(t) and c1?(t)trace out the
pathshown to the right
Note that the total areain this new basis is
still p/4!
Area of a circle of radius ½.
Hmm, is this true for any basis? Yes!

a p/4
c0?
c1?
26
Action Some Terminology

A physical action is, most generally, the
integral of an energy over time.
Or, along some temporalizable path.
Typical units of action h or ?.
Correspond to angles of 1 circle and 1 radian,
respectively.
Normally, the word action is reserved to refer
specifically to the action of the Lagrangian L.
This is the action in Hamiltons least action
principle.
However, note that we can broaden our usage a bit
and equally well speak of the action of any
quantity that has units of energy.
E.g., the action of the Hamiltonian H L pv
L p2/m.
Warning I will use the word action in this
more generalized sense!

27
Action as Computation

We will argue Action is computation.
That is, an amount of action corresponds exactly
to an amount of physical quantum-computational
work.
Defined in an appropriate sense.
The type of action corresponds to the type of
computational work performed, e.g.,
Action of the Hamiltonian All computational
work.
Action of the Lagrangian Internal
computational work.
Action of pv Motional computational work
We will show exactly what we mean by all this,
mathematically

28
Action of the Hamiltonian

Consider now the action A (eq. (1) below) of any
time-dependent Hamiltonian operator H(t).
Note that A is an Hermitian observable as well.
The H determines state-vector dynamics via the
usual Schrödinger relation d/dt iH/?.
For our purposes, we are adopting the opposite of
the usual (but arbitrary) sign convention in this
equation.
This leads to the time-evolution operator (2)
below
Given H(t) ? A(t0,t) ? U(t0,t), any initial
vector v(t0) yields a complete state trajectory
v(t) U(t,t0)v(t0).

(1)
(2)
29
Some Nice Identities for A

Consider applying a specific operator A itself to
any initial state v0.
For any observable A, well use shorthand like
Av0 ?v0Av0?.
It is easy to show that Av0 is equal to all of
the following
The quantum-average net phase-angle accumulation
of the coefficients ci of vs components in Hs
energy eigenbasis vi, weighted by the component
probabilities (3).
The line integral, along vs trajectory, of the
magnitude of the imaginary part of the inner
product ?v v dv ? between adjacent states
(4).
Exactly twice the net area a swept out in the
complex plane (relative to the complex origin) by
vs coefficients cj, in any basis vj.
We will prove this.
Note that the value of Av0 therefore depends only
on the specific trajectory v(t) that is taken by
v0 itself,
and not on any other properties of the complete
Hamiltonian that was used to implement that
trajectory!
For example, it doesnt depend on the energy Hu
assigned to other states u that are orthogonal to
v.

(4)
(3)
30
Area swept out in energy basis

For a constant Hamiltonian,
By a coefficient ci ofan energy basisvector vi.
If rici1, the area swept out is ½ of the
accumulatedphase angle.
For rilt1, note areais this times ri2.
Sum over i ½ avg.phase angle accumulated ½
action of Hamiltonian.

ri
31
In other bases

Both the phase and magnitude of each coefficient
will change, in general
- The area swept out is no longer just a
corresponding fraction of a circular disc.
Its not immediately obvious that
the sum of the areas swept out by all the
cjs will still be the same in the new
basis.
- Well show that indeed it is.

32
Basis-Independence of a

Note that each cj(t) trajectory is just a sum of
circular motions
Namely, a linear superposition of the ci(t)
motions
Since each circular component motion is
continuous and differentiable, so is their sum.
The trajectory is everywhere a smooth curve.
No sharp, undifferentiable corners.
Thus, in the limit of arbitrarily short time
intervals, the path can always be treated as
linear.
Area daj approaches ½ the parallelogram area rj
rj' sin d? cjcj'
Cross product of complex numbers considered as
vectors
Use a handy complex identity ab ab i(ab)
Implies that daj ½ Imcj cj'
So, da ½ Imvv'.
So da is basis-independent, since the inner
product vv' is!

33
Computational Work of a Hamiltonian applied to a
system

Suppose were given a time-dependent Hamiltonian
H(t), a specific initial state v, and a time
interval (t0, t)
We can of course compute the operator A(t0,t)
from H.
Well call Av the computational work performed
according to the specific action operator A (or
by H acting from t0 to t) on the initial state
v.
Later we will see some reasons why this
identification makes sense.
For now, take it as a definition of what we mean
by computational work
If we are given only a set V of possible initial
vectors,
The (maximum, minimum) work of A (or H from t0 to
t) is (5)
If we had a prob. dist. over V (or equiv., a
mixed state ?),
we could instead discuss the expected work (6) of
A acting on V

(5)
(6)
34
Computational Effort to Cause a Desired Change

If we are interested in taking v0 to v1, and we
have a set ? of available action operators A
(implied, perhaps, by a set of available
Hamiltonians H(t))
we define the minimum work or effort to get
from v0 to v1, (7)
Maximizing over ? isnt very meaningful, since it
may often yield 8.
And if we have a desired unitary transform U that
we wish to perform on any of a set of vectors V,
given a set ? of available action operators,
Then we can define the minimum (over ?)
worst-case (over V) work to perform U, or
worst-case effort to do U (8).
Similarly, we can discuss the best-case effort to
do U.
or (if we have vector probabilities) the minimum
(over ?) expected (over V) work to do U, or
expected effort to do U (9).

(8)
(7)
(9)
35
The Justification for All This

Why do we insist on referring to these concepts
as computational work or computational
effort?
One could imagine other possible terms, such as
amount of change, physical effort, the
original action of the Hamiltonian etc.
What is so gosh-darned computational about this
concept?
Answer We can use these concepts to quantify the
size or difficulty of, say, quantum
logic-gate operations.
And by extension, classical reversible operations
embedded in quantum operations
And by extension, classical irreversible Boolean
ops, embedded within classical reversible gates
with disposable ancillas
As well as larger computations composed from such
primitives.
The difficulty of a given computational op
(considered as a unitary U) is given by its
effort (minimized work over ?)
We can meaningfully discuss an operations
minimum, maximum, or expected effort over a given
space of possible input states.

36
But, you say, Hamiltonian energy is only defined
up to an additive constant

Still, the effort of a given U can be a
well-defined (and non-negative) quantity, IF
We adopt an appropriate and meaningful zero of
energy!
One useful convention
Define the least eigenvalue (ground state energy)
of H to be 0.
This ensures that energies are always positive.
However, we might want to do something different
than this in some cases
E.g., if the ground-state energy varies, and it
includes energy that had to be explicitly
transferred in from another subsystem
Another possible convention
We could count total gravitating mass-energy
Anyway, lets agree, at least, to just always
make sure that all energies are positive, OK?
Then the action is always positive, and we dont
have to worry about trying to make sense of a
negative amount of computational work.

37
Energy as Computing

Given that Action is computation,
That is, amount of computation,
where the suffix -ation denotes a noun,
i.e., the act itself,
What, now, is energy?
Answer Energy is computing.
By which I mean, rate of computing activity.
The suffix -ing denotes a verb,
the (temporal) carrying out of an action
This should be clear, since H(t) dA/dt
Thus the Hamiltonian energy of any given state is
the rate at which computational work is being (or
would be) performed on (or by, if you prefer)
that state.

38
Applications of the Concept

How is all this useful?
It lets us calculate time/energy tradeoffs for
performing operations of interest.
It can help us find (or define) lower bounds on
the number of operations of a given type needed
to carry out a desired computation.
It can tell us that a given implementation of
some computation is optimal.

39
Time/Energy Tradeoffs

Suppose you determine that the effort of a
desired v1?v2 or U(V) (given the available
actions ?) is A.
For a multi-element state set V, this could be a
minimum, maximum, or expected effort
And, suppose the energy that is available to
invest in the system in question is at most E.
This then tells you directly that the
minimum/maximum/expected (resp.) time to perform
the desired transformation will be t A/E.
To achieve equality might require varying the
energy of the state over time, if the optimal
available H(t) says to do so
Conversely, suppose we wish to perform a
transformation in at most time t.
This then immediately sets a scale-factor for the
magnitude of the energy E that must be devoted to
the system in carrying out the optimal
Hamiltonian trajectory H(t) i.e., E A/t.

40
Single-Qubit Gate Scenario

Lets first look at 2-state (1-qubit) systems.
Later well consider larger systems.
Let U be any unitary operator in U2.
I.e., any arbitrary 1-qubit quantum logic gate.
Let the vector set V consist of the sphere of
all unit vectors in the Hilbert space H2.
Given this scenario, the minimum effort to do any
U is always 0 (just let v be an eigenvector of
U), and is therefore uninteresting.
Instead well consider the maximum effort.
What about our space ? of available action
operators?
Suppose for now, for simplicity, that all
time-dependent Hermitian operators on H2 are
available as Hamiltonians.
Really we only need the time-independent ones,
however.
Thus, ? consists of all (constant) Hermitian
operators.

41
Analysis of Maximum Effort

The maximum effort to do U (in this scenario)
arises from considering a geodesic trajectory
in U2.
All the worst-case state vectors just follow the
most direct path along the unit sphere in
Hilbert space to get to their destinations.
Other vectors go along for the ride on the
necessary rotation.
The optimal unitary trajectory U(t0,t) then
amounts to a continuous rotation of the Bloch
sphere around a certain axis in 3-space
where the poles of the rotation axis are the
eigenvectors of U.
Also, theres a simultaneous (commuting) global
phase-rotation.
If we also adopt the convention that the
ground-state energy of H is defined to be 0,
Then the global phase-rotation factor goes away,
And we are left with a total effort A that turns
out to be exactly equal to ??, where 0 ? p is
simply the (minimum) required angle of
Bloch-sphere rotation to implement the given U.

42
Some Special Cases

Pauli operators X,Y,Z (including XNOT), as well
as the Hadamard gate
Bloch sphere rotation angle p (rads)
Maximum effort h/2
Square-root of NOT, also phase gate (square root
of Z)
Rotation angle p/2, effort h/4.
p/8 gate (square root of phase gate)
Rotation angle p/4, effort h/8.

43
Fidelity and Infidelity

The fidelity between pure states u,v is defined
as F(u,v) ?uv?.
So, F2 is the probability of conflating the two.
Define the infidelity between u,v as
Thus, I2 1 - F2 is the probability that if
state u is measured in a basis that includes v as
a basis vector, it will project to a basis state
other than v.
Infidelity is thus a distance metric between
states

44
Effort Required for Infidelity

Guess what, a Bloch-sphere rotation by an angle
of ? gives a maximum (over V) infidelity of I(?)
sin(2?).
Meanwhile, the minimum fidelity is cos(2?)
Youll notice that F2I21, as probabilities
should.
Therefore, achieving an infidelity of I requires
performing a U whose maximum effort is at least A
2?arcsin(I).
However, the specific initial states that
actually achieve this infidelity under the
optimal rotation are Bloch equator states
Equal superpositions of high and low energy
eigenstates
They perform a quantum-average amount of
computational work that is only half of the
maximum effort.
Thus, the actual work required for an infidelity
of I is only half of the maximum effort, or W
A/2 ?arcsin(I).
And so, a specific state that carries out an
amount of computational work W p/2 can achieve
an infidelity of at most I sin(W/?), while
maintaining a fidelity of at least Fcos(W/?)
a nice simple relation Especially if we let ?1

45
Multi-qubit Gates

Some multi-qubit gates are easy to analyze
E.g., controlled-U gates that perform a unitary
U on one qubit only when all of the other qubits
are 1
If the space of Hamiltonians is truly totally
unconstrained, then (it seems) the effort of
these will match that of the corresponding 1-bit
gates.
However, in reality we dont have such
fine-tailored Hamiltonians readily available.
A more thorough analysis would analyze the effort
in terms of a Hamiltonian thats expressible as a
sum of realistically-available, 1- and 2-qubit
controllable interaction terms.
We havent tried to do this yet

46
Conclusion

We can define a clear and stable measure of the
length of any continuous state trajectory in
Hilbert space. (Call it computational work.)
Its simply given by the action of the
Hamiltonian.
It has a nice geometric interpretation as well.
From this, we can accordingly define the size
(or effort) of any unitary transformation.
As the worst-case (or average-case) path length,
minimized over the available Hamiltonians.
We can begin to quantify the effort required for
various quantum gates of interest
From this, we can compute lower bounds on the
time to implement them for states of given energy.