Convex decomposition of dimension-altering quantum channels

Wang, Dongsheng

doi:10.1142/s0219749916500453

Cited by 7 publications

(25 citation statements)

References 32 publications

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“…Methods for implementing quantum channels from a system A to a system B with low experimental cost as a sequence of simple-to-perform operations were considered in [3,4,13,14]. In [3], a lower bound on the number of parameters required for a quantum circuit topology that is able to perform arbitrary extreme channels from m to n qubits (i.e., from a system A of dimension d A = 2 m to a system B of dimension d B = 2 n ) was given.…”

Section: Application: Implementation Of Quantum Channelsmentioning

confidence: 99%

Smooth manifold structure for extreme channels

Iten

Colbeck

2018

Journal of Mathematical Physics

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A quantum channel from a system A of dimension dA to a system B of dimension dB is a completely positive trace-preserving map from complex dA × dA to dB × dB matrices, and the set of all such maps with Kraus rank r has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the rdB ×dA dimensional Stiefel manifold. Secondly, as the set of all Choi-states of a fixed rank r. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system A to system B of a fixed Kraus rank r is a smooth submanifold of dimension 2rdAdB − d 2 A − r 2 of the set of all Choi-states of rank r. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from A to B arbitrarily well.

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Section: Application: Implementation Of Quantum Channelsmentioning

confidence: 99%

Smooth manifold structure for extreme channels

Iten

Colbeck

2018

Journal of Mathematical Physics

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“…We also note that Ansätze for decompositions of arbitrary channels have been considered in [22,23]. One of these, Ansatz 1 in [23], is based on applying the Cosine-Sine decomposition to the Stinespring dilation isometry of the channel, and hence will always work [16]. Our results imply that the Ansatz given in [22] (which is designed for the RandomQCM) cannot work in general because it does not have enough parameters.…”

Section: Introductionmentioning

confidence: 97%

“…In the special case of a single-qubit channel, we recover a circuit topology (consisting of only one C-not gate) similar to that given in [21]. We also note that Ansätze for decompositions of arbitrary channels have been considered in [22,23]. One of these, Ansatz 1 in [23], is based on applying the Cosine-Sine decomposition to the Stinespring dilation isometry of the channel, and hence will always work [16].…”

Section: Introductionmentioning

confidence: 99%

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Quantum circuits for quantum channels

2017

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We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of Controlled-not (C-not) gates required (single-qubit gates are free). We consider three different models. In the first, the Quantum Circuit Model (QCM), we consider sequences of single-qubit and C-not gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of C-not gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from m qubits to n qubits that uses at most one ancilla and has a low C-not count. We give explicit examples for small numbers of qubits that provide the lowest known C-not counts.

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“…Given the importance of Choi states, we develop a method for their preparation by quantum circuits based on extreme channels [15,16], which is more efficient than the standard dilation method. A mixed program state will be stored as a set of extreme program states and random bits, and the later can greatly reduce the circuit costs for preparing program states.…”

Section: Introductionmentioning

confidence: 99%

Choi states, symmetry-based quantum gate teleportation, and stored-program quantum computing

Wang

2019

Preprint

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The stored-program architecture is canonical in classical computing, while its power has not been fully recognized for the quantum case. We study quantum information processing with stored quantum program states, i.e., using qubits instead of bits to encode quantum operations. We develop a stored-program model based on Choi states, following from channel-state duality, and a symmetry-based generalization of deterministic gate teleportation. Our model enriches the family of universal models for quantum computing, and can also be employed for tasks including quantum simulation and communication.

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Convex decomposition of dimension-altering quantum channels

Cited by 7 publications

References 32 publications

Smooth manifold structure for extreme channels

Smooth manifold structure for extreme channels

Quantum circuits for quantum channels

Choi states, symmetry-based quantum gate teleportation, and stored-program quantum computing

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