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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 separabilitypreserving operations.
Separable states
•
TFAE:
r is separable (r2Sep).
r=k 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.
–
–
–
•
•
Gates @ states
r(E) ´ (EAB1A’B’) (|FiAA’|FiBB’)
Alice
A
B
A0
B0
|FiAA’
E
Bob
|FiBB’
r(E) + local operations can probabilistically simulate E
[Cirac et al]
Separable operations
TFAE:
1.
2.
3.
4.
E is a separable quantum operation.
E(s) = k(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=k (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 roundoff 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).
Part 2: Tools.
How much noise makes a gate
separable?
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=j aj |ji|ji.
• Rs(|yi)=Rg(|yi) = (j aj)2-1.
• Rr(|yi)=d2a1a2.
Schmidt decomposition of
unitary gates
• Any unitary gate U can be decomposed as
U = lk Ak Bk, with k |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=k lk Ak Bk with AkAky=BkBky=I/d,
then Rs(U)=Rg(U)=Rs(r(U))=(k lk)2-1.
• For example, Rg(CNOT)=1. The optimal
noise process is a classical CNOT.
Part 3: Results
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 separabilitypreserving operations.
• Determine how much entangling power is
necessary for computation.
Simulating separabilitypreserving 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).