Post on 20-Dec-2015
Gate robustness:
How much noise will ruin a quantum gate?
Aram Harrow and Michael Nielsen, quant-ph/0212???
Outline1. 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.
Separable states
• TFAE:– is separable (2Sep).
– =k pk |kihk| |kihk|
– 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.
Gates states
(E) ´ (EAB1A’B’) (|iAA’|iBB’)
A
A0
|iAA’
B
B0
|iBB’
E
(E) + local operations can probabilistically simulate E [Cirac et al]
Alice Bob
Separable operations
TFAE:1. E is a separable quantum operation.
2. E() = k(AkBk)(AkyBk
y)
3. (E1)Sep ½Sep (E cannot create entanglement)
4. (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=k (Ak Bk)¢(Aky Bk
y) to |1i|2i.
• Let |ki=Ak|1i Bk|2i and pk=hk|ki.• We obtain pk
-1/2|ki 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() = 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(||) = min R such that +R is separable.
• Random robustness: Rr() = R(||I/d).
• Separable robustness: Rs()=minR(||) where is separable.
• General robustness: Rg()=minR(||).
• Rg() · Rs() · Rr().
Robustness of pure states (q-ph/9806094)
• Suppose |i=j aj |ji|ji.
• Rs(|i)=Rg(|i) = (j aj)2-1.
• Rr(|i)=d2a1a2.
Schmidt decomposition of unitary gates
• Any unitary gate U can be decomposed as U = k Ak Bk, with k |k|2=1 and TrAjAk
y=TrBjBky=djk.
• The Schmidt coefficients of (U) are {k}.
• Thus Rr(U)=Rr((U))=d412.
• For qubits (d=2), Rr(U)· Rr(CNOT)=8.
“Unital” gates.
• If U=k k Ak Bk with AkAky=BkBk
y=I/d, then Rs(U)=Rg(U)=Rs((U))=(k k)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 in time T poly log(1/).