Detrimental Decoherence

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DetrimentalDecoherence

• Gil Kalai
• Hebrew University of Jerusalem
• And Yale University
• QEC07, Los Angeles, Dec 07
• HU quantum computing sem. Jan.08

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Prepared for QEC07

• First International Conference on Quantum Error
  Correction
• University of Southern California,
• Los Angeles 17-21 December, 2007.

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Revised for HU quantum computer seminar

• Hebrew University of Jerusalem Thursdays
  quantum computation seminar, January 17, 2008

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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.

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• BACKGROUND

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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.

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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 0gt b 1gt
• in a two-dimensional complex space Uq. We
  can think of the qubit q as representing '0' with
  probability a2 and '1' with probability b2.

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Quantum Computers (cont.)

• The state of the entire computer is a unit vector
  in the 2n dimensional tensor product of these
  vector spaces Uqs for the individual qubits.
• The state of the computer thus represents a
  probability distribution on the 2n strings of
  length n of 0s and 1s.

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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 2k- dimensional 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.

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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.

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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.

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Quantum Computation (cont.)

• Peter Shor (1994) proved that factoring an
  n-digits 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.

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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.

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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.

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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.

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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).

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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).

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Noisy Quantum Computation (cont.)

• Noisy quantum computers in every computer-cycle
  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.
  

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Quantum Error Correction

• The second major development (Shor, Steane, 1995)
  towards fault-tolerant quantum computation was
  the discovery of quantum error correction codes.

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The threshold theorem

• Finally, the threshold theorem (1997
  Aharonov-BenOr, Kitaev, Knill-Laflamme-Zuerk)
  asserts that when the noise rate is small, and
  the noise is local, fault tolerant quantum
  computation (FTQC) is possible.

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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.

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• Daniel Gottesmans picture is worth thousand
  words

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The Classical and Quantum Worlds Daniel Gottesman
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• This lecture deals with the desert of
  decoherence.
• In this desert quantum processes are modelled
  by unprotected quantum circuits.

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• Examples first
• Unprotected quantum circuits and a simple type of
  errors.

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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.

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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
  error-propagation 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.

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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.

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Our main thesis

• Quantum noisy systems are best modeled by
  unprotected quantum circuits.

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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 S D(x)Ex .

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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 ks.)

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• Conjectures
• On
• Decoherence
• For noisy quantum computers

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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.

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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 error-synchronization.

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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.

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C Censorship

• Conjecture C The states of noisy quantum
  computers are approximately local.

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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.

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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 gt 0, and for
  every subspace U of V the infinitesimal rate of
  noise restricted to U is at least
• K log (dim U).

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• Detrimental Decoherence
• For noisy quantum computers
• Conjectures A,B,C.

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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.

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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 .

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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

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Conjecture A (cont.)

• For a state T of the quantum computer, a standard
  measure of entanglement is the mutual information
  
• S(ab) S(Ta ) S (Tb ) S(T a,b)
• (S is the entropy function.)
• The formal version of conjecture A is
• L(a,b) gt K(L(a),L(b)) S(ab)
• For general form of errors the formal definition
  of L(a,b) is more complicated but the basic idea
  is similar.

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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.

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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.

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A stronger formulation III Many qubits

• Another strong form of conjecture A applies to
  larger sets of qubits.

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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 (ta) qubits are damaged is
  exponentially small with the number of qubits n
  for every agt0.

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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.

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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.

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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!

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Mathematical challenges (cont.)

• It would also be nice to have an entropy based
  description of error-synchronization without
  referring to the expansion in terms of
  tensor-product of Pauli operators.

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• An extension to
• general quantum systems

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• 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.

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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.

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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.

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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
  non-equilibrium thermodynamics.

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• Models

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Models

• Models exhibiting conjectures A and B should
  exhibit them already for the storage-errors (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.

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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.

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• Consequences of Detrimental Decoherence
• Computational complexity

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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
  quasi-polynomial overhead) classical computations
  and log-depth quantum computation.
• Log-depth quantum computation ( classical
  computation) is good enough for polynomial-time
  factoring.

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Aaronsons 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.

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• The rate of errors
• And decoherence free subspaces

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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.

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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.

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Critique of the tentative conjecture

• Conjecture E (v. 1) is quite problematic. If
  QEC fails we can indeed expect (as the effect of
  errorpropagation) 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. Lets try
  another avenue.

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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 Kgt0, for
  every subspace U of V, the infinitesimal rate of
  noise restricted to U is at least
• K log (dim U).

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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 log-depth quantum
  computation.

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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 Kgt0,
  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.

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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
  non-commutativity 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.

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• Difficulties and potential counter examples
• A few difficulties and potential counterexamples
  for conjectures A, B and C are described.

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Two photons

• Errors for two far-away entangled photons are not
  correlated.
• (So the rate of detrimental errors in this case
  is 0.)

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Classical fault tolerance

• If fault tolerant quantum computing fails, how is
  it that fault tolerance classical computing
  prevails?

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• 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.

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Superconductivity

• Is superconductivity a counter example?
• (Or, at least, isnt it true that similar
  pessimistic conjectures could have been raised
  regarding superconductivity had it not been
  witnessed?)

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2n bosons

• (This is a potential counter example I cooked by
  myself.) A state of 2n bosons each having a
  ground state 0gt and an excited state 1gt 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.)

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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

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• Conclusion
• The story we try to tell

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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.

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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...

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Clarkes 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.

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Anyway, it is fun. Thank you!