1. Gate robustness: How much noise will ruin a quantum gate? Aram Harrow and Michael Nielsen, quant-ph/0212???

2. Outline 1. Why do we care? –Separable operations cannot create entanglement. –A classical computer can efficiently simulate a circuit composed of separable * operations. 2. How do we solve it? –The state-gate isomorphism (Choi/Jamiolkowski). –State robustness (Vidal and Tarrach, q-ph/9806094) 3. Do we have any results? –Upper bounds on the accuracy threshold. –The CNOT is the most robust two-qubit gate. –Depolarizing noise is hardest to correct.

3. Part 1: Motivation. Separable and separability- preserving operations.

4. Separable states TFAE: –  is separable (  2 Sep). –  =  k p k |  k ih  k | ­ |  k ih  k | –  can be created with local operations and shared randomness. Sep may be useful for quantum computing. Sep can be used for non-classical tasks, such as data hiding states.

5. Gates  states  ( E ) ´ ( E AB ­ 1 A ’ B ’ ) (|  i AA’ ­ |  i BB’ ) A A0A0 |  i AA’ B B0B0 |  i BB’ E  ( E ) + local operations can probabilistically simulate E [Cirac et al] AliceBob

6. Separable operations TFAE: 1. E is a separable quantum operation. 2. E (  ) =  k (A k ­ B k )  (A k y ­ B k y ) 3.( E­ 1)Sep ½ Sep ( E cannot create entanglement) 4.  ( E ) 2 Sep. Note: LOCC ( {separable operations} (e.g. decoding data hiding states)

7. Separability-preserving operations E is separability-preserving if E¢ Sep ½ Sep. Example: SWAP is separability-preserving. Question: Is {separability-preserving operations on n parties} = Hull{ E±P : E is separable and P is a permutation}? Claim: A quantum circuit comprised of separable operations can be simulated efficiently on a classical computer.

8. Classical simulation algorithm Suppose we apply E =  k (A k ­ B k ) ¢ (A k y ­ B k y ) to |  1 i­ |  2 i. Let |  k i =A k |  1 i­ B k |  2 i and p k = h  k |  k i. We obtain p k -1/2 |  k i with probability p k. If we use b bits of precision, then the round- off error is 2 -b p k 1/2. Since k=1,…,16, it is very unlikely that we obtain a very small p k (or a very large p k -1/2 ).

9. Part 2: Tools. How much noise makes a gate separable?

10. Gate robustness Robustness: R( E || F ) = min R such that E +R F is separable. Random robustness: R r ( E ) = R( E || D ) where D (  ) = I/d. Separable robustness: R s ( E )=min F R( E || F ) where F is separable. General robustness: R g ( E )=min F R( E || F ). R g ( E ) · R s ( E ) · R r ( E ).

11. State robustness (Vidal & Tarrach, 9806094) Robustness: R(  ||  ) = min R such that  +R  is separable. Random robustness: R r (  ) = R(  ||I/d). Separable robustness: R s (  )=min  R(  ||  ) where  is separable. General robustness: R g (  )=min  R(  ||  ). R g (  ) · R s (  ) · R r (  ).

12. Robustness of pure states (q-ph/9806094) Suppose |  i =  j a j |j i |j i. R s (|  i )=R g (|  i ) = (  j a j ) 2 -1. R r (|  i )=d 2 a 1 a 2.

13. Schmidt decomposition of unitary gates Any unitary gate U can be decomposed as U = k A k ­ B k, with  k | k | 2 =1 and TrA j A k y =TrB j B k y =d  jk. The Schmidt coefficients of  (U) are { k }. Thus R r (U)=R r (  (U))=d 4 1 2. For qubits (d=2), R r (U) · R r (CNOT)=8.

14. “Unital” gates. If U=  k k A k ­ B k with A k A k y =B k B k y =I/d, then R s (U)=R g (U)=R s (  (U))=(  k k ) 2 -1. For example, R g (CNOT)=1. The optimal noise process is a classical CNOT.

15. Part 3: Results

16. The threshold theorem For arbitrary two-qubit gates subject to independent depolarizing noise, the threshold is p th <(8- p 8)/7 ¼ 0.74. Different models give different bounds on the threshold.

17. Optimal gates vs. optimal noise processes R r (U) is maximized for the CNOT, with R r (U) · R r (CNOT)=8 for all two-qubit gates. Conversely, the completely depolarizing channel, D, is the most effective noise process against arbitrary gates: min E max U R(U|| E )=max U R(U|| D )=d 4 /2.

18. Goals Tighter bounds on the threshold. General formulas for R s (U) and R g (U). Characterize the set of separability- preserving operations. Determine how much entangling power is necessary for computation.

19. Simulating separability- preserving gates Theorem: Let C be a quantum circuit involving only separability-preserving gates and single-qubit measurements. If C uses T gates, then there exists a classical algorithm that can reproduce the measurement statistics of C to accuracy  in time T poly log(1/  ).