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Detrimental
Decoherence
Gil Kalai
Hebrew University of Jerusalem
And Yale University
QEC07, Los Angeles, Dec ’07
HU quantum computing sem. Jan.‘08
Prepared for
QEC07
First International Conference on
Quantum Error Correction
University of Southern California,
Los Angeles 17-21 December, 2007.

Revised for HU quantum
computer seminar
Hebrew University of Jerusalem – Thursday’s
quantum computation seminar, January
17, 2008
Outline of the talk
1. Quantum computers, noisy quantum
computation and fault tolerance. Examples.
2. Detrimental decoherence: conjectures
3. Extensions and models
4. The rate of errors.
5. Comments on classical noise, computational
complexity, possible counterexamples, etc.
BACKGROUND
Quantum Computers
Quantum computers (Deutsch, 85) are
hypothetical devices based on quantum
physics. Here is a brief description of what
they are:
The state of a digital computer having n bits is
a string of length n of zeros and ones. As a
first step towards quantum computers we can
consider (abstractly) stochastic versions of
digital computers where the state is a
(classical) probability distribution on all such
strings.
Quantum Computers (cont.)
Quantum computers are similar to these
(hypothetical)
stochastic classical computers
and they work on qubits (say n of them).
The state of a single qubit q is described by a
unit vector
u = a |0> + b |1>
in a two-dimensional complex space U[q]. We
can think of the qubit q as representing '0'
with probability |a|2 and '1' with probability
|b|2.
Quantum Computers (cont.)
The state of the entire computer is a unit
vector in the 2n dimensional tensor
product of these vector spaces U[q]s
for the individual qubits.
The state of the computer thus
represents a probability distribution on
the 2n strings of length n of 0’s and 1’s.
Quantum Computers (cont.)
The evolution of the quantum computer is
via ``gates.'' Each gate G operates on k
qubits, and we can even assume that k
equals one or two. Every such gate
represents a unitary operator on the 2kdimensional tensor product of the
spaces that correspond to these k
qubits.
In every cycle of the computer, gates act
in parallel on disjoint sets of qubits.
Quantum Computers (cont.)
Moving from a qubit q at a certain state to
the probability distribution it
represents is called a measurement.
We can assume that measurements of the
qubits that amount to a sampling of 0-1
strings according to the distribution
these qubits represent, is the final step
of the computation.
Quantum Computation (BQP)
Quantum computers as described earlier, (or
according to quite a few alternative but
computationally equivalent descriptions,) are
capable of doing everything classical
computers do and more. The remarkable
complexity class described by polynomial
time quantum computation is called BQP.
Quantum Computation (cont.)
Peter Shor (1994) proved that factoring an ndigits number has a polynomial time quantum
algorithm, hence is in BQP.
There is evidence that BQP goes well beyond
factoring and that NP-complete problems
are much beyond BQP.
Are quantum computers feasible?
The feasibility of (computationally superior)
quantum computers is one of the most
exciting (and clear-cut) scientific problems
of our time.
If feasible, QC may represent an amazing
new physics reality based on human
technology. QC being unfeasible may
represent quite surprising new insights in
physics theory.
Related issues to QC feasibility
The feasibility of quantum computers is also
relevant to other issues of considerable
interest that arose independently (and even
earlier). Here is a partial list:
1) The evolution of open quantum systems.
2) The “measurement problem” and other
issues in the foundations of quantum
mechanics.
Related issues to QC feasibility
(cont.)
3. The existence of (stable) non abelian
anyons.
4. Thermodynamics, non-equilibrium
thermodynamics. (Suggestions for “4th law of
thermodynamics”, “superthermal particles”, etc.)
5. Noise.
The Postulate of Noise
An early critique of quantum computers put
forward in the mid-90s by Landauer, Unruh,
and others concerned the matter of noise:
The postulate of noise: Quantum
systems are noisy.
Understanding the meaning and nature of noise
(and the reason for noise) is of great
importance in this context (as in many
others).
Noisy Quantum Computation
Dealing with the issue of noise required
three important developments: The
first was a formal development of a
model of noisy quantum computation.
This was first carried out by Bernstein
and Vazirani (1993).
Noisy Quantum Computation
(cont.)
Noisy quantum computers: in every computercycle there are some “storage errors” which
describe a certain deterioration of the state
of the computer compared to its intended
state. In addition, the gates are not perfect
and this is expressed by “gate errors”. Of
course, these two types of errors propagate
along the computation.
Quantum Error Correction
The second major development (Shor,
Steane, 1995) towards fault-tolerant
quantum computation was the discovery
of quantum error correction codes.
The threshold theorem
Finally, the threshold theorem (1997;
Aharonov-BenOr, Kitaev, KnillLaflamme-Zuerk) asserts that when the
“noise rate” is small, and the noise is
“local”, fault tolerant quantum
computation (FTQC) is possible.
Detrimental errors
Detrimental errors are hypothetical forms
of errors for noisy quantum computers
(and more general open quantum
systems) which are damaging for
quantum error-correction and quantum
fault-tolerance.
Detrimental errors for quantum
computers and their effects are
described by three conjectures and are
discussed in this lecture.
Daniel Gottesman’s
picture is worth
thousand words
The
Classical
and
Quantum
Worlds
Daniel
Gottesman
This lecture deals with
the “desert of
decoherence”.
In this “desert”
quantum processes
are modelled by
“unprotected
quantum circuits”.
Examples first:
Unprotected
quantum circuits
and a simple type
of errors.
Unprotected quantum
programs
An important example to have in mind is error-propagation
of unprotected quantum programs or circuits.
Take the standard model of independent errors and
suppose that the error rate is so small that it
accumulates at the end of the computation to a small
constant-rate error. This was first studied by Unruh.
For such errors we will witness that rather than being
independent the errors will tend to synchronize.
Unprotected quantum
programs –words of caution
Since the error-propagation of unprotected
quantum circuits serves as a “role model” for a
damaging noise, it is tempting to regard errorpropagation as the sort of damaging noise for
QEC.
This is not the case! Whatever bad properties we
would like to consider they should be
manifested already for the “new” errors in each
computer cycle. When the new errors behave
nicely, FTQC deals well with their propagation.
Unprotected quantum
programs –Cavaet 2
Following Unruh we take the standard model of
independent errors and suppose that the error
rate is so small that it accumulates at the end
of the computation to a small constant-rate
error.
We conjecture that the incremental (new) errors
themselves behave like the acccumulation of
errors in an unprotected circuits. This also
means that taking small rate errors according to
the standard noise models is only a first
approximation to the behavior of unprotected
quantum circuits.
Our main thesis
Quantum noisy systems are
best modeled by
unprotected quantum
circuits.
A simple class of errors
Let Wk represent the error of changing
the kth qubit to the fixed state of
maximum entropy. For a 0-1 string x of
length n let Ex denote the tensor product
of error operations: Wk when xk = 1 and
the identity Ik when xk = 0.
For a probability distribution D on all 0-1
strings of length n let ED = Σ D(x)Ex .
An even simpler class of errors
For most of the lecture we can consider
just errors of the form ED . We will
mention now an even smaller class. Let w
be a probability distribution on the unit
interval [0,1]. We can define a probability
distribution D(w) on 0-1 strings of length
n in two steps as follows: First we choose
t in [0,1] according to w and then we let
every xk =1 with probability t
(independently for different k’s.)
Conjectures
On
Decoherence
For noisy quantum
computers
A: Information leaks for pairs
of qubits
Conjecture [A]: A noisy quantum computer is
subject to error with the property that
information leaks for two substantially entangled
qubits have a substantial positive correlation.
Conjecture [A] refers to part of the overall errors
affecting noisy quantum computers. But we
conjecture that the effect of detrimental
errors (described by Conjectures [B] and [C])
cannot be remedied by errors of a different
type.
B: Error Synchronization
Error-synchronization refers to a situation
where, while the error rate is small, there
is a substantial probability of errors
affecting a large fraction of qubit.
Conjecture [B]: For any noisy quantum
computer at a highly entangled state
there will be a strong effect of errorsynchronization.
Approximately-local states
A (pure) state of a quantum computer is
approximately local if it is determined
(up to a small error) by the induced
states of small sets of qubits.
Note that this is a combinatorial and not
a geometric notion. Note also that
states needed for quantum (many-)
error corrections are not approximately
local.
C: Censorship
Conjecture [C]: The states of noisy
quantum computers are approximately
local.
D: An extension
A proposed extension of detrimental
errors to general quantum systems reads:
Conjecture [D]: A description (or
prescription) of a noisy quantum system
at a state S is subject to error described
by a quantum operation E that tends to
commute with every unitary operator that
stabilizes S.
E: The rate of errors
Trying to understand the rate of
detrimental errors leads to:
Conjecture [E]: Any noisy quantum system
whose states are described by a Hilbert
space V is subject to noise so that for
some K > 0, and for every subspace U of
V the infinitesimal rate of noise
restricted to U is at least
K log (dim U).
Detrimental
Decoherence
For noisy quantum
computers:
Conjectures [A],[B],[C].
The setting
As described before, we consider a noisy
quantum computer whose “intended” state is
pure, and we assume that along the evolution
the overall error, namely the gap between the
ideal state and the actual state is small.
The errors can be described by a unitary
operator on the computer qubits and the
“neighborhood qubits” or as a quantum
operation E on the space of density matrices
for these n qubits.
The setting (cont.)
The errors we consider are the “new
errors” in a single computer cycle.
In the discussion of conjectures [A] and
[B] we assume for simplicity that the
errors are of the form ED .
Conjecture [A]
Remember that we restrict ourselves to
errors of the form ED which depend on a
probability distribution on 0-1 strings of
length n. The error rate L(a) for the kth
qubit a is simply the probability that xk
=1. If b is the jth qubit, let L(a,b) be
the correlation between the event xk =1
and the event xj = 1
Conjecture [A] (cont.)
For a state T of the quantum computer, a
standard measure of entanglement is the
mutual information
S(a;b) = S(T|a ) + S (T|b ) – S(T| {a,b})
(S is the entropy function.)
The formal version of conjecture [A] is:
L(a,b) > K(L(a),L(b)) S(a;b)
For general form of errors the formal
definition of L(a,b) is more complicated but
the basic idea is similar.
A stronger formulation I:
Two qudits
Conjecture [A] extends to pairs of qudits
rather than pairs of qubits without
change.
In this generality it applies to disjoint sets
of qubits in a noisy quantum computer.
A stronger formulation II:
Emergent entanglement
Entanglement between qubits can emerge
when we measure other qubits and “look
at” the results. A strong form of
conjecture [A] takes this into account
and replaces entanglement with a more
general notion of emergent
entanglement.
A stronger formulation III:
Many qubits
Another strong form of conjecture [A]
applies to larger sets of qubits.
B: Error Synchronization
Suppose that the error rate for every
qubit is t. For our error models ED this
means that the probability that xk =1 is
t for every k.
In the standard models of noise the
probability that a fraction of (t+a)
qubits are damaged is exponentially
small with the number of qubits n for
every a>0.
B: Error Synchronization
Error synchronization means that for some t
which is much larger than s there is a
substantial probability that
xk =1 for t or more indices k.
For example, when w is a probability distribution
on [0,1] and we consider the distribution ED(w) .
The standard models of noise assume that w is
a Dirac distribution (supported on one point).
We will witness error synchronization if the
average of w is t but w is supported on much
larger real numbers.
Error Synchronization?
An aside: Is error synchronization something we can really
expect in highly correlated systems? Is this something
we witness in nature? Two quick remarks:
a) Perhaps we do see error-synchronization even in
correlated classical systems.
b) The hoped-for-argument would be counterfactual.
Highly entangled systems as required in quantum
computers (will lead to) come along with very strong
error synchronization (that we do not often encounter),
which in turn implies that such highly entangled states
are unrealistic.
Conjecture C and Mathematical
challenges
For lack of time we will not attempt to
describe formally conjecture C. Once
described mathematically a remaining
challenge will be to deduce conjectures
[B] and [C] from conjecture [A] and its
extensions. Errors of the form ED can
serve as a good starting point. We would
also like to deduce from the
conjectures on physical qubits similar
statements for protected qubits!
Mathematical challenges (cont.)
It would also be nice to have an entropy
based description of errorsynchronization without referring to
the expansion in terms of tensorproduct of Pauli operators.
An extension to
general quantum
systems
If the conjectures we propose are correct
they should represent a property of
noise which is not limited to quantum
computers.
However our conjectures [A], [B] and [C]
strongly rely on the tensor product
structure of the Hilbert space
describing the states of quantum
computers.
Conjecture [D]
Conjecture [D]: A description (or
prescription) of a noisy quantum system
at a state S is subject to error
described by a quantum operation E
that tends to commute with every
unitary operator that stabilizes S.
Conjecture [D]: why and what
The rationale behind [D] goes as follows:
Our conjectures suggest that if E
represents the error for state S and E'
represents the error for state U(S),
for a unitary operator U on V, then E'
will be ``close'' to U-1EU. In particular,
this implies that if U(S)=S then E' is
``close'' to U-1EU; hence UE is
``close'' to EU.
Conjecture [D]: why and what
(cont.)
Greg Kuperberg pointed out that at a
thermodynamics equilibrium a certain
limiting error E will actually commute
with every U that stabilizes S. One
possible way to regard Conjecture [D] is
as a statement referring to nonequilibrium thermodynamics.
Models
Models
Models exhibiting conjectures [A] and [B]
should exhibit them already for the storageerrors (or gate-errors). The new errors may
be represented by a rapid quantum circuit.
Such models may be created by pushing the
model of Aharonov, Kitaev and Preskill a little
further. Error synchronization arises in a
paper by Klesse and Frank.
Here is a toy model that can be examined.
A toy model
There are no gate errors. Consider the
graph G whose vertices are the qubits
and whose edges are qubits that occur in
a gate. Edges are labeled by the gate
imperfection.
The storage error is described by ED
where the probability distribution D is
given by an Ising model on the graph G
based on these gate-imperfections.
Consequences of
Detrimental
Decoherence:
Computational
complexity
How damaging are low rate
detrimental errors
I would expect that detrimental errors will fail
current methods for fault tolerance and
quantum linear error correction.
On the other hand, low rate detrimental errors
may still allow (with polynomial or quasipolynomial overhead) classical computations
and log-depth quantum computation.
Log-depth quantum computation (+ classical
computation) is good enough for polynomialtime factoring.
Aaronson’s Shor/sure challenge
Scott Aaronson suggested a very nice challenge:
Propose a restriction on QC that will not allow
polynomial time factoring and would not
violate empirical results.
This looks very difficult. I am not aware of
methods that will allow a reduction to a
computational power below log-depth quantum
computing, when the error-rate is small.
The rate of errors
And decoherence
free subspaces
High-rate errors
A major obstacle for fault tolerance is
high error-rate.
When we consider the standard models
and perceptions regarding noise there
is not much reason to believe that the
error rate (for individual qubits) will
increase in terms of the number of
qubits of the computer.
If we examine unprotected quantum
circuits things are different.
The rate of errors for
unprotected quantum circuits
For unprotected quantum circuits, not only
do the errors tend to synchronize, but the
error-propagation causes the error-rate
itself to depend on the complexity of the
target state. This may suggest a tentative
conjecture:
Conjecture [E] (v.1) The rate of detrimental
errors in a noisy quantum computer is
higher for highly entangled states.
Critique of the tentative
conjecture
Conjecture [E] (v. 1) is quite problematic. If QEC
fails we can indeed expect (as the effect of
error–propagation) that the error rate will
increase when we prepare complicated states.
However, as is, this conjecture adds little more
to the conjecture “QEC fails”. Moreover,
unlike conjectures [A] and [B], where both the
assumptions and conclusions depended on the
tensor product structure, here the conclusion
does not depend on this structure. Let’s try
another avenue.
Rate of errors – take 2
The common convention about the rate of noise
is that in every computer cycle there is a
positive small probability for every qubit to
be damaged. The infinitesimal rate of errors
for k qubits taken together is just k times
that of a single qubit error-rate.
Conjecture [E] (v.2) : Any noisy quantum system
whose states are described by a Hilbert
space V is subject to noise so that for some
K>0, for every subspace U of V, the
infinitesimal rate of noise restricted to U is
at least
K log (dim U).
Rate of errors – take 2 (cont.)
This (very strong and rather general)
conjecture [E] can be regarded as a
formulation of the postulate of noise
that runs directly against the idea of
decoherence-free subspaces. It agrees
with the behavior we observe for
unprotected quantum circuits.
Conjecture [E] may damage even logdepth quantum computation.
Conjecture [E] (cont.)
Conjecture [E] (repeated): Any noisy quantum system
whose states are described by a Hilbert space V is
subject to noise so that for some K>0, for every
subspace U of V the infinitesimal rate of noise
restricted to U is at least
K log (dim U).
In order to exclude decoherence free subspaces,
Conjecture [E] would imply error-synchronization.
Moreover, the rate (for a single qubit) of highly
synchronized errors will scale up linearly with the
number of qubits.
The rate of errors (cont.)
We can also expect that the rate K of
detrimental errors for a prescribed (or
described) evolution of a quantum
system, depends on a measure of noncommutativity between the space P of
unitary operators leading to the state
from the initial state, and the space F
of unitary operators leading from the
state to the terminal state.
Difficulties and
potential counter
examples
A few difficulties and potential counterexamples for
conjectures [A], [B] and [C] are described.
Two photons
Errors for two far-away entangled
photons are not correlated.
(So the rate of detrimental errors in this
case is 0.)
Classical fault tolerance
If fault tolerant quantum computing fails,
how is it that fault tolerance classical
computing prevails?
The formal versions (and wordings) of the
conjectures are “tailored” to avoid
these two difficulties.
Still these are genuine difficulties that
should be kept in mind.
Superconductivity
Is superconductivity a counter example?
(Or, at least, isn’t it true that similar
pessimistic conjectures could have been
raised regarding superconductivity had it not
been witnessed?)
2n bosons
(This is a potential counter example I
cooked by myself.) A state of 2n bosons
each having a ground state |0> and an
excited state |1> so that each state has
occupation number precisely n appears
to violate Conjecture C. Is it
realistic? (If the occupation number has
a normal distribution this is OK.)
nonabelyons
Stable non abelian anyons, which some expect to
witness rather soon, run against our
conjectures.
(There is much theoretical and empirical effort
regarding creation, detection and applications
of non abelyan anyons. I am not aware of a
systematic theoretic study for why they
cannot be created.)
Fermi  fermions
Bose  bosons
Any  anyons
Conclusion:
The story we try to tell
Conclusion
We are trying to describe a story of our physical
world without quantum error-correction,
decoherence-free subspaces and perhaps even
without quantum computing which goes beyond
classical computing. (But, of course, a story well
within quantum mechanics.)
We start telling it in a very special way - just about
two qubits ([A]) so that it could be tested easily
for small devices. But we also tried to tell it in a
very general way ([D] and [E]) which goes beyond
quantum computers.
Conclusion (cont.)
We try to tell the story as formally and as
explicitly as possible (this makes for most of the
effort and there is a way to go), and to make it
quantitative. We tried to make our story bold as
to make it easy to refute. ([C] and [E] are the
boldest. Does [E] violate the empirical results
presented by Laflamme?) We point out
surprising aspects (Error-synchronization [B])
and we consider some analogies (classical noise).
We attempt to make it into an elegant story.
Of course, at the end it also has to be correct...
Clarke’s three laws of prediction
1) When a distinguished but elderly scientist
states that something is possible, he is almost
certainly right. When he states that something
is impossible, he is very probably wrong.
2) The only way of discovering the limits of the
possible is to venture a little way past them into
the impossible.
3) Any sufficiently advanced technology is
indistinguishable from magic.
Anyway, it is fun. Thank you!