INTERNET DRAFT M. Ohta
draft-ohta-qec-inapplicable-00.txt Tokyo Institute of Technology
Intended status: Informational October 30, 2020
Expires: May 3, 2021
Quantum Error Correction Inapplicable to Really Entangled States
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Abstract
Though quantum error correction assumes localized error model of Shor
that errors on a qubit are caused by interaction with its local
environment, enabling essentially classical error correction for
unentangled states, the model is applied to entangled states
improperly without involving local environment states in the
entanglement.
That is, when an entangled state (Q) is represented as superposition
of unentangled terms (Qi) as Q=Q1+Q2+...+Qn, local environment states
around qubits are, in general, different term by term. Q will be,
with term-specific error operators (Ei), E1*Q1+E2*Q2+...+En*Qn, not,
with a common error operator (E) assumed by Shor, E*(Q1+Q2+...+Qn).
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A complication is that Shor's error model is a little quantum,
allowing for two different local environment states around a qubit.
As such, quantum error correction is applicable to some trivially
entangled states including states used by Shor code but not to really
entangled states.
1. Introduction
An assumption of noise model for quantum error correction by Shor [1]
is "The critical assumption here is that decoherence only affects one
qubit of our superposition, while the other qubits remain unchanged.
It is not clear how reasonable this assumption is physically, but it
corresponds to the assumption in classical information theory of the
independence of noise.", which means a qubit suffers from error as a
result of interaction with local environment around the qubit but no
interaction occurs with other qubits or local environment of other
qubits. Though some extension to consider certain interaction
between a qubit and other qubits or environment of other qubits is
possible, some locality is still assumed.
The error model is directly applicable to unentangled, that is,
essentially classical, states, resulting in localized errors,
corrections of which are essentially classical error correction.
However, it is unreasonable to expect such localized errors for
entangled states, because the states themselves do not have locality.
Actually, with a 2 qubit entangled state: |00>+|11>, if the first
qubit coherently interacts with its environment to be |0>, the entire
state becomes |00>, which means the second qubit is also affected,
Though the case is trivial enough to be explained by Shor's error
model as superposition of identity (no error) and sign flip (|0> and
|1> become |0> and -|1>, correspondingly) error:
|00>=((|00>+|11>)+(|00>-|11>))/2, such an explanation dose not deny
lack of locality of errors on entangled states.
As Shor overlooked the fact that when qubit states are entangled,
their environment states are, in general, also entangled, errors on
really entangled states are highly non-local to which quantum error
correction is not applicable.
That is, when an entangled state (Q) is represented as superposition
of (minimum number of) unentangled terms (Qi) as Q=Q1+Q2+...+Qn,
local environment states around a qubit are, in general, involved in
the entanglement and different term by term, resulting in different
error operators (Ei). As a result, Q will be disturbed by noise to be
E1*Q1+E2*Q2+...+En*Qn, whereas, Shor thought a common error operator
(E) is applicable to all the terms as E*(Q1+Q2+...+Qn).
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It is obvious that, with some clever encoding using fixed number of
extra qubits, effect of E may be compensated, which was quantum error
correction, but the extra qubits are not enough to compensate all the
Ei's (with quantum algorithms, 'n' will often be exponentially large
w.r.t. problem size).
A complication is that Shor's error model is a little quantum,
allowing for, seemingly despite his intention, two different local
environment states around a qubit, which is explained in the next
section.
2. Why Shor's Error Model is a little Quantum?
In [1], Shor explicitly described environment state of a qubit before
interaction with the qubit |e0> (same state for |0> and |1>, which
should be the intention of Shor) and described interaction
(decoherence) process as:
|e0>|0> -> |a0>|0>+|a1>|1>
|e0>|1> -> |a2>|0>+|a3>|1>
where |a0>, |a1>, |a2> and |a3> are environment states after the
interaction. |a0>, |a1>, |a2> and |a3> are "not generally orthogonal
or normalized" [1] and can be fully independent each other. Ignoring
error terms,
|e0>|0> -> |a0>|0>
|e0>|1> -> |a3>|1>
So, if qubit state is |0>, its environment state is |a0>, but, if
qubit state is |1>, its environment state us |a3>, different from
|a0>.
It should also be noted that, as |a0>, |a1>, |a2> and |a3> are fully
independent each other, the process may have two different initial
environment states as:
|e0>|0> -> |a0>|0>+|a1>|1>
|e1>|1> -> |a2>|0>+|a3>|1>
So, Shor's error model is slightly quantum allowing for different
environment states depending on qubit values.
As such, errors on trivially entangled states (e.g., superposition of
just two unentangled states) such as |00>+|11> and
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(|000>+|111>)(|000>+|111>)(|000>+|111>) should be correctable. As
the latter example is Shor code for |0>, experimental confirmation of
Shor's quantum error correction should success, as long as the input
qubit to an error correction circuit is unentangled with other qubits
outside of the circuit, which is not the case when quantum algorithms
are run on quantum computers relying on aggressive entanglement
between qubits.
It should be noted that, though it does not affect the points of this
memo, Shor's representation of qubit and its environment states using
tensor product is inappropriate, because, for the interaction,
relative phase between them matters (e.g., resulting states of
homodyne detection relies on the relative phase), which can be
represented by not tensor but Cartesian product. Though |e0>|0> and
-|e0>|0> represent a same state, (|e0>, |0>) and (|e0>, -|0>) are
different states.
It should also be noted that Shor's error model is a little quantum
not because sign flip error is quantum specific and classically
impossible. It is merely that sign flip error does not occur on
modern computers where phase is not used to encode information. In
an optical packet router using FDLs (Fiber Delay Lines) as optical
buffers (memory), like ancient computers with Mercury delay lines as
memory, where QAM (Quadrature Amplitude Modulated) PDM (Polarization
Division Multiplexed) signal is sent over the FDLs [2], sign flip
errors occur as relative phase errors between polarization modes.
3. Conclusions
It is shown that not-really-quantum error correction works only for
errors with mostly classical locality and is not applicable to non-
local errors on really entangled states.
As qubit states within quantum computers running quantum algorithms
are really entangled, quantum error correction for them is
impossible, which makes construction of quantum computers with
practical size practically impossible.
Entangled states, in general, are a lot noisier than Shor thought,
which should be the reason why the states are so fragile easily
collapsing to be less noisy less entangled or unentangled states.
4. Security Considerations
That construction of quantum computers with practical size is
practically impossible means quantum computers do not make public key
cryptography unsafe, though there may still be some classical
algorithm to make it unsafe.
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5. IANA Considerations
This memo has no actions for IANA.
Informative References
[1] P. W. Shor, "Scheme for reducing decoherence in quantum computer
memory", Phys. Rev. A, Oct. 1995,
http://www.cs.miami.edu/~burt/learning/Csc670.052/pR2493_1.pdf.
[2] M. Ohta, "Optical switching of many wavelength packets: A
conservative approach for an energy efficient exascale
interconnection network", 2016 IEEE 17th International Conference on
High Performance Switching and Routing (HPSR),
https://ieeexplore.ieee.org/document/7525641, August 2016.
Author's Address
Masataka Ohta
Tokyo Institute of Technology
2-12-1-W8-54, O-okayama, Meguro-ku
Tokyo 152-8552
JAPAN
Phone: +81-3-5734-3299
Fax: +81-3-5734-3299
EMail: mohta@necom830.hpcl.titech.ac.jp
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