Gate robustness:

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: • r is separable (r2Sep). • r=åk pk |akihak| ­ |bkihbk| • r 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. E A B Alice Bob A0 B0 |FiAA’ |FiBB’ Gates @ states r(E) ´ (EAB­1A’B’) (|FiAA’­|FiBB’) r(E) + local operations can probabilistically simulate E [Cirac et al]

6. Separable operations TFAE: • E is a separable quantum operation. • E(s) = åk(Ak­Bk)s(Aky­Bky) • (E­1)Sep ½ Sep (E cannot create entanglement) • r(E)2Sep. Note: LOCC ( {separable operations} (e.g. decoding data hiding states)

7. Separability-preserving operations • E is separability-preserving if E¢Sep½Sep. • 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=åk (Ak­ Bk)¢(Aky­ Bky) to |y1i­|y2i. • Let |fki=Ak|y1i­ Bk|y2i and pk=hfk|fki. • We obtain pk-1/2|fki with probability pk. • If we use b bits of precision, then the round-off error is 2-bpk1/2. Since k=1,…,16, it is very unlikely that we obtain a very small pk (or a very large pk-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+RF is separable. • Random robustness: Rr(E) = R(E||D) where D(r) = I/d. • Separable robustness: Rs(E)=minFR(E||F) where F is separable. • General robustness: Rg(E)=minFR(E||F). • Rg(E) · Rs(E) · Rr(E).

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

12. Robustness of pure states (q-ph/9806094) • Suppose |yi=åj aj |ji|ji. • Rs(|yi)=Rg(|yi) = (åj aj)2-1. • Rr(|yi)=d2a1a2.

13. Schmidt decomposition of unitary gates • Any unitary gate U can be decomposed as U = lk Ak­ Bk, with åk |lk|2=1 and TrAjAky=TrBjBky=ddjk. • The Schmidt coefficients of r(U) are {lk}. • Thus Rr(U)=Rr(r(U))=d4l1l2. • For qubits (d=2), Rr(U)· Rr(CNOT)=8.

14. “Unital” gates. • If U=åklk Ak­ Bk with AkAky=BkBky=I/d, then Rs(U)=Rg(U)=Rs(r(U))=(åklk)2-1. • For example, Rg(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 pth<(8-p8)/7¼0.74. • Different models give different bounds on the threshold.

17. Optimal gates vs. optimal noise processes • Rr(U) is maximized for the CNOT, with Rr(U)· Rr(CNOT)=8 for all two-qubit gates. • Conversely, the completely depolarizing channel, D, is the most effective noise process against arbitrary gates: minE maxU R(U||E)=maxU R(U||D)=d4/2.

18. Goals • Tighter bounds on the threshold. • General formulas for Rs(U) and Rg(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 e in time T poly log(1/e).