Approximate quantum error correction for

correlated noiseAvraham Ben-Aroya

Amnon Ta-ShmaTel-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.

In this talk

We ask whether errors that are• Highly correlated• Restrictedcan be approximately corrected.

Specifically: we study noise on a single qubit that is controlled by all other qubits.

Controlled qubit flip Ei,S for i [n], S{0,1}n-1 define the error

Extend linearly.5

Operator In S? Basis vector

X on the i’th qubit yes 000000

I no 000001

X on the i’th qubit yes 000010

…. … ….

I no 111110

I no 111111

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.

It’s 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.

Approximate quantum ECC

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

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

= αk |k

’ = Ei,S() = kSXi(αk |k ) + kSαk |k

D(’) = kSαk |kSynd(Xi)+ kSαk |kSynd(I) 13

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-|11114

Why the Majority function?

Notice that Ei,S (α |x1 x2 … xi … xn +β |x1 x2 … ,xi 1,… xn )

either α |x +β |x ei

or β |x +α |x ei

Thus, it is invariant if α= βi.e. f(x)= f(xei )

A non-trivial code for Ei,S

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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Ii(f)=Prx [f(x) ≠f(x ei)]

A high dimensional code for Ei,S

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Product

f f f

Block 1,0 Block 1,0 Block b,0Block b,0

f

Z1 Zb

fz1..zb(x1,0,…,xb,0,xb,1)=i f(xi,zi)

x 1,0 x 1,1 x b,0 x b,1x 1,0 x b,1

Z1=0 Zb=1

f(x1,0) f(xb,1)

Idea: Take many independent, low influence functions

A negative result

Controlled phase-flips cannot be corrected. For S1…S4={0,1}n define:

ES1,…,S4|v = eik|v for vSk

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 idea1. 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 ei|i

’= r’i e’i|i

3. Conclude that and ’ have a high inner product (>0.1). 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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