Gate robustness:

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Gate robustness:. How much noise will ruin a quantum gate?. Aram Harrow and Michael Nielsen, quant-ph/0212???. Outline. 1. Why do we care? Separable operations cannot create entanglement. A classical computer can efficiently simulate a circuit composed of separable * operations.

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### Gate robustness:

How much noise will ruin a quantum gate?

Aram Harrow and Michael Nielsen, quant-ph/0212???

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.

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

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.

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]

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)

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

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.
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).