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On Almost Randomizing Channels with a Short

Kraus Decomposition, Guillaume Aubrun

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

For large d, we study quantum channels on C d obtained by selecting ran- domly N independent Kraus operators according to a probability measure µ on the unitary group U(d). When µ is the Haar measure, we show that for N d/e 2 , such a channel is e-randomizing with high probability, which means that it maps every state within distance e/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discreti- zation argument. Moreover, for general µ, we obtain an e-randomizing channel provided N d (log d) 6 /e 2 .F ord = 2 k (k qubits), this includes Kraus operators obtained by ten- soring k random Pauli matrices. This leads to more efficient constructions of almost randomizing channels. The proof uses recent results on empirical processes in Banach spaces.

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For large d, we study quantum channels on C d obtained by selecting ran- domly N independent Kraus operators according to a probability measure µ on the unitary group U(d). When µ is the Haar measure, we show that for N d/e 2 , such a channel is e-randomizing with high probability, which means that it maps every state within distance e/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discreti- zation argument. Moreover, for general µ, we obtain an e-randomizing channel provided N d (log d) 6 /e 2 .F ord = 2 k (k qubits), this includes Kraus operators obtained by ten- soring k random Pauli matrices. This leads to more efficient constructions of almost randomizing channels. The proof uses recent results on empirical processes in Banach spaces.
The concept of $\e$-randomizing quantum channels has been introduced by Hayden, Leung, Shor and Winter in connection with approximately encrypting quantum states. They proved using a discretization argument that sets of roughly dlog⁡dd \log ddlogd random unitary operators provide examples of such channels on $\C^d$. We show that a simple trick improves the efficiency of the argument and reduces the number of unitary operators to roughly ddd.
Randomization of quantum states is the quantum analogue of the classical one‐time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d/ε2) Pauli operators to map any d‐dimensional state to a state that is within trace distance ε of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter, and Ambainis and Smith.Then, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d‐dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small ε.
Abstract. In this study, we generate quantum channels with random Kraus operators to typically obtain almost twirling quantum channels and quantum expanders. To prove the concentration phenomena, we use matrix Bernstein’s inequality. In this way, our random models do not utilize Haar-distributed unitary matrices or Gaussian matrices. Rather, as in the preceding research, we use unitary t-designs to generate mixed tensor-product unitary channels acting on Cdt. Although our bounds in Schatten p-norm are valid only for 1 ≤p≤ 2, we show that they are typically quantum ϵ-twirling channels with the tail bound proportional to 1/poly(dt), while such bounds were previously constants. The number of required Kraus operators was also improved by powers of logdandt, to be proportional todtlogd/ϵ2. Such random quantum channels are also typically quantum expanders, but the number of Kraus operators must grow proportionally totlogdin our case. Finally, a new non-unital model of super-operators generated by bounded and isotropic random Kraus operators was introduced, which can be typically rectified to yield almost randomizing quantum channels and quantum expanders.
Abstract In this work we investigate how quantum expanders (i.e., quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a 222-design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form U⊗kU^{\otimes k}U⊗k, with UUU sampled from a 2k2k2k-design measure, then the corresponding random quantum channel is typically an optimal kkk-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).
We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space A to states on some output Hilbert space B can be compressed into one with order dlog⁡d Kraus operators, where d=max(|A|,|B|) , hence much less than |A||B| . In the case where the channel's outputs are all very mixed, this can be improved to order d . We discuss the optimality of this result as well as some consequences.
The random purification channel maps n copies of any mixed quantum state to n copies of a random purification of the state. We generalize this construction to arbitrary symmetries: for any group G of unitaries, we construct a quantum channel that maps states contained in the algebra generated by G to random purifications obtained by twirling over G. In addition to giving a surprisingly concise proof of the original random purification theorem, our result implies the existence of fermionic and bosonic Gaussian purification channels. As applications, we obtain the first tomography protocol for fermionic Gaussian states that scales optimally with the number of modes and the error, as well as an improved property test for this class of states.