1. Approximate quantum error correction for correlated noise
Avraham Ben-Aroya
Amnon Ta-Shma
Tel-Aviv University

2. 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.

3. 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.

4. 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.

5. Controlled qubit flip Ei,S for i [n], S{0,1}n-1 define the error
Operator
In S?
Basis vector
X on the i’th qubit
yes
000000 I no
000001
000010 …. …
111110
111111
Extend linearly. 5 5

6. 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?

7. 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.

8. 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. 8

9. 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.

10. 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.

11. 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()

12. Syndrome decoding If {Ei} is a set of errors that we allow, and,
Assume, we have decoding D s.t.
D(Ei ) =   Synd(Ei)

13. 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() = kSXi(αk |k ) + kSαk |k
D(’) = kSαk |kSynd(Xi)+ kSαk |kSynd(I)  

14. 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

15. 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= β
i.e. f(x)= f(xei )

16. 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. 16

17. 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(x1,0) f f f
f(xb,1)
Block 1,0
x 1,0
x 1,0
Block 1,0
x 1,1
Block b,0
x b,0
x b,1
x b,1
Block b,0
Z1=0 Z1 Zb
Zb=1 17

18. ES1,…,S4|v = eik|v for vSk
A negative result
Controlled phase-flips cannot be corrected.
For S1…S4={0,1}n define:
ES1,…,S4|v = eik|v 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. 18

19. 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 19

20. 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i|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. 20

21. Open questions What kind of errors can be approximately corrected?
Under which errors can we achieve fault-tolerant computation? 21