Approximate quantum error correction for correlated noise

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Approximate quantum error correction for
correlated noise

• Avraham Ben-Aroya
• Amnon Ta-Shma
• Tel-Aviv University

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The standard quantum noise model

• Allowed error any combination of noise
  operators that act on at most t qubits.
• There are QECC of length n that can correct ?(n)
  errors.

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How many errors?

• No QECC can of length correct n/4 errors.
• Crepeau, Gottesman, Smith
• An approximate QECC that can correct about
  n/2 errors. (some restrictions apply).
• Approximate ECC may be much more powerful than
  perfect ECC.

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In this talk

• We ask whether errors that are
• Highly correlated
• Restricted
• can be approximately corrected.
• Specifically we study noise on a single qubit
• that is controlled by all other qubits.

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Controlled qubit flip

• Ei,S for i? n, S?0,1n-1 define the error

Extend linearly.
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Our results

• A positive result controlled single bit flip
• Cannot be quantumly corrected
• Can be approximately corrected
• A negative result controlled phase flips
• Cannot be approximately corrected
• Natural question what can be approximately
  corrected?

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Motivation I

• We have a good understanding of what can be
  perfectly corrected.
• We do not have such an understanding for
  approximate correction.
• Its a natural question.

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Motivation II

• Quantum ECC and quantum fault tolerance are basic
  tools for constructing quantum computers that can
  withstand noise.
• It is not clear at all what is the true noise
  model that affects a quantum computer. The answer
  probably depends on the actual realization.
• It makes sense to study which errors can and
  cannot be approximately corrected.

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Our work

• Is just a first step.
• It deals with a toy example.
• But it already gives a negative result.
• We hope it will stimulate further research.

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Approximate quantum ECC
We require that the decoded state is close to
the original codeword.

• A code C ?-corrects a family of errors ?, if
  there is a POVM, D, such that ???C ?E?? D(E?)
  has 1-? fidelity with ?.
• Almost error free subspaces a special kind of
  approximate QECC where the decoding procedure is
  simply the identity.

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Controlled qubit flip cannot be corrected

• Thm A QECC that corrects Ei,S i?n, S?n
  has at most one codeword.
• Proof Based on the characterization that a code
  C corrects a family of errors ? iff
• ??,??C ?E1,E2 ?? ??? ? E1(?)?E2(?)

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Syndrome decoding

• If Ei is a set of errors that we allow, and,
• Assume, we have decoding D s.t.
• D(Ei ?) ? ? Synd(Ei)

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The problem with ctrl qubit flip errors

• Ei,S flips the i-th qubit for basis vectors in S
• It acts differently on different basis vectors
• ? ? ak k?
• ? Ei,S(?) ?k?SXi(ak k? ) ?k?Sak k?
• D(?) ?k?Sak k??Synd(Xi) ?k?Sak k??Synd(I) ?
  ?

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A non-trivial code for Ei,S

• The code is spanned by two codewords.
• The two basis codewords
• ??k k?
• ? ?k f(k) k?
• With f being the Majority function.
• ? 000?001?010?011?100?101?
  110?111?
• ? 000?001?010?-011?100?-101?-
  110?-111?

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Why the Majority function?

• Notice that
• Ei,S (a x1 x2 xi xn? ß x1 x2 ,xi ?1,
  xn? )
• either
• a x? ß x ?ei?
• or
• ß x? a x ?ei?
• Thus, it is invariant if a ß
• i.e. f(x) f(x?ei )

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A non-trivial code for Ei,S
Ii(f)Prx f(x) ?f(x ? ei)

• Thm the code O(1/?n) corrects Ei,S.
• Proof
• We prove For any codeword ? ,
• ?Ei,s ?- ?? ? I(f) ??
• Thus, any function with low influence (like
  Majority or Tribes) is good.

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A high dimensional code for Ei,S
Idea Take many independent, low influence
functions
Product
fz1..zb(x1,0,,xb,0,xb,1)?i f(xi,zi)
f
f
f
f
f(x1,0)
f(xb,1)
Block 1,0
Block 1,0
Block b,0
Block b,0
x 1,0
x 1,1
x b,0
x b,1
x 1,0
x b,1
Z1
Zb
Z10
Zb1
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A negative result

• Controlled phase-flips cannot be corrected.
• For S1??S40,1n define
• ES1,,S4v? ei?kv? for v?Sk
• ? 1 0, ? 2 ?/2, ? 3 ?, ? 4 3?/2
• Thm A QECC that 0.1-corrects the class of errors
  defined above has at most one codeword.

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A negative result Proof idea

• 1. In any vector space of dimension 2 , there are
  two-codewords ?,? such that the inner product of
  their magnitudes is big.
• ? ? ai i?
• ? ? bi i?
• ? ai bi 1/2

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A negative result Proof idea

• 2. Use the controlled phase errors to make the
  phase of the two vectors close to each other.
• ? ? ri e?ii?
• ? ? ri e?ii?
• 3. Conclude that ? and ? have a high inner
  product (gt0.1). Thereofre there is no way to
  correct this error.

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Open questions

• What kind of errors can be approximately
  corrected?
• Under which errors can we achieve fault-tolerant
  computation?

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Controlled bit flip Classical setting

• In the classical error model we allow the
• adversary to look at the actual codeword and
• then decide which t bits to flip.
• The noise may depend on the codeword!

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Controlled bit flip - classical setting
Several codewords
0001111
1000110
1011010
Noise
Adversary that flips at most one bit
0001110
1000110
1011011
Error correction
A classical ECC can easily correct a single bit
flip
0001111
1000110
1011010
The error may depend on the codeword!
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Controlled qubit flip
viv1,,vi-1,vi1,..vn

• For i? n, S?0,1n-1 define the error
• (I?I?X?I?I) v? If vi ?
  S
• Ei,S v?
• v?
  Otherwise

Highly correlated noise! All qubits together
determine whether the i-th qubit is flipped.
and extend linearly. Defined with respect to the
standard basis.
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Two classical noise models

• Shannon Stochastic noise.
• E.g., every bit is flipped with independent
• probability p
• Hamming Adversarial
• The adversary looks at the code word yC(x)
• and decides which t bits to flip.