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

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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.

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

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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.

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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

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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)

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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.

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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 ).

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

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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 ).

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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 ( ).

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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.

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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.

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“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.

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Part 3: Results

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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.

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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.

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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.

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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/ ).

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