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

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. 2

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 E i,S for i  [n], S  {0,1} n-1 define the error Extend linearly. 5 OperatorIn S?Basis vector X on the i’th qubityes000000 Ino000001 X on the i’th qubityes000010 ….… Ino111110 Ino111111

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? 6

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. 7

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

11. Controlled qubit flip cannot be corrected Thm: A QECC that corrects {E i,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  E 1,E 2   :   E 1 (  )  E 2 (  ) 11

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

13. The problem with ctrl qubit flip errors  E i,S flips the i-th qubit for basis vectors in S  It acts differently on different basis vectors  =  α k |k   ’ = E i,S (  ) =  k  S X i (α k |k  ) +  k  S α k |k  D(  ’) =  k  S α k |k  Synd(X i )+  k  S α k |k  Synd(I)   13

14. A non-trivial code for E i,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  14

15. Why the Majority function? Notice that E i,S (α |x 1 x 2 … x i … x n  +β |x 1 x 2 …,x i  1,… x n  ) either α |x  +β |x  e i  or β |x  +α |x  e i  Thus, it is invariant if α= β i.e. f(x)= f(x  e i )

16. A non-trivial code for E i,S Thm: the code O(1/  n) corrects {E i,S }. Proof:  We prove: For any codeword , |  * E i,s  -  *  |  I(f) |  *  |  Thus, any function with low influence (like Majority or Tribes) is good. 16 I i (f)=Pr x [f(x) ≠f(x  e i )]

17. A high dimensional code for E i,S 17 Product ff f Block 1,0 Block b,0 f Z1Z1 ZbZb f z1..zb (x 1,0,…,x b,0,x b,1 )=  i f(x i,z i ) x 1,0 x 1,1 x b,0 x b,1 x 1,0 x b,1 Z 1 =0Z b =1 f(x 1,0 ) f(x b,1 ) Idea: Take many independent, low influence functions

18. A negative result  Controlled phase-flips cannot be corrected. For S 1  …  S 4 ={0,1} n define: E S 1,…,S 4 |v  = e i  k |v  for v  S k  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.  =  a i |i   =  b i |i   | a i b i | ≥ 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.  =  r i 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

22. 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! 22

23. Controlled bit flip - classical setting 0001111 Several codewords: 10001101011010 Adversary that flips at most one bit 000111010001101011011 A classical ECC can easily correct a single bit flip 000111110001101011010 Noise Error correction The error may depend on the codeword! 23

24. Controlled qubit flip For i  [n], S  {0,1} n-1 define the error (I  …I  X  I…  I) |v  If v i  S E i,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. v i =v 1,…,v i-1,v i+1,..v n 24

25. 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 y=C(x) and decides which t bits to flip.