Compatible transformations for a qudit decoherence-free/noiseless encoding

Bishop, C. Allen; Byrd, Mark

doi:10.1088/1751-8113/42/5/055301

Cited by 13 publications

(13 citation statements)

References 65 publications

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“…From these operators, we can understand how the information of the data qubit is distributed in the encoded state. We note that direct operations on logical qubits in DFS/NS were discussed in [26].…”

Section: Figmentioning

confidence: 99%

Implementation of a simple operator-quantum-error-correction scheme

2013

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It is often assumed that the ancilla qubits required for encoding a qubit in quantum error correction (QEC) have to be in pure states, |00 . . . 0 for example. In this letter, we introduce an encoding scheme avoiding fully correlated errors, in which the ancillae may be in a uniformly mixed state. We demonstrate our scheme experimentally by making use of a three-qubit NMR quantum computer. Moreover, the encoded state has an interesting nature in terms of Quantum Discord, or purely quantum correlations between the data-qubit and the ancillae.A quantum computer is vulnerable against environmental noise and it must be protected by one way or another. Quantum error correction (QEC) is one of the most successful approaches to this end [1]. Despite this great success, QEC requires expensive resources, or ancillae that are usually assumed to be in pure states [2,3]. However, it is not yet proved that ancillae in uniformly mixed states are useless. We extend previous works [4] and show an encoding scheme robust against fully correlated noise in which all the ancillae can be in uniformly mixed states. The encoded state has an interesting nature in terms of Quantum Discord [5], or purely quantum correlations between the data-qubit and the ancillae. Our QEC scheme also provides an example of Deterministic Quantum Computation with 1-Qubit (DQC-1) [6,7].Suppose we have a single qubit in an arbitrary state ρ 1 , which we want to protect from noise. We introduce some additional qubits (ancillae) in order to protect the first qubit and suppose that all the qubits suffer from the same noise. Such a noise is called fully correlated and it may happen when the dimensions of the quantum computer are microscopic compared with the wavelength of external disturbances. Noiseless subsystem (NS) [8][9][10][11] and decoherence free subsystem (DFS) [12][13][14][15] are well known strategies to protect a system from such fully correlated noises [16,17]. These schemes, however, require ancillae in pure states and thus they are expensive.In the following, we show that it is indeed possible to devise a cheaper QEC scheme employing ancillae in the uniformly mixed state. Letbe the state of the qubit to be protected. Here σ 0 is a unit matrix of dimension 2, n = (n x , n y , n z ) is the Bloch vector, and σ i is the ith component of the Pauli matrices. We introduce two ancillae in uniformly mixed states, whose Bloch vectors are 0. The initial state of the three-qubit system is thus a tensor product state ρ 1 ⊗ (σ 0 /2) ⊗2 . The unitary encoding operator U E transforms the tensor product state to an entangled stateρ 3 . If the state of the system is againρ 3 even after the action of noises, a unitary recovery operator, U R = U † E , transforms ρ 3 back to the initial tensor product state ρ 1 ⊗ (σ 0 /2) ⊗2 and ρ 1 can be recovered after tracing over the ancilla states.It is highly counterintuitive that a QEC scheme works with ancillae in uniformly mixed states. The trick is that the uniformly mixed state (σ 0 /2) ⊗2 is rewritten aswhere n 2 and n ′ ...

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Section: Figmentioning

confidence: 99%

Implementation of a simple operator-quantum-error-correction scheme

2013

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“…(2),we see that the two bases may be converted from one to another using, for example, the correspodence between the raising and lower operators of the group and the hermitian basis. To be specific, one would expand ρ = r,s ρ rs E rs , and [29] and references therein. )…”

Section: Affine Map From the Dynamical Mapmentioning

confidence: 99%

General open-system quantum evolution in terms of affine maps of the polarization vector

Byrd

Bishop

2011

Phys. Rev. A

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The operator-sum decomposition (OS) of a mapping from one density matrix to another has many applications in quantum information science. To this mapping there corresponds an affine map which provides a geometric description of the density matrix in terms of the polarization vector representation. This has been thoroughly explored for qubits since the components of the polarization vector are measurable quantities (corresponding to expectation values of Hermitian operators) and also because it enables the description of map domains geometrically. Here we extend the OS-affine map correspondence to qudits, briefly discuss general properties of the map, the form for particular important cases, and provide several explicit results for qutrit maps. We use the affine map and a singular-value-like decomposition, to find positivity constraints that provide a symmetry for small polarization vector magnitudes (states which are closer to the maximally mixed state) which is broken as the polarization vector increases in magnitude (a state becomes more pure). The dependence of this symmetry on the magnitude of the polarization vector implies the polar decomposition of the map can not be used as it can for the qubit case. However, it still leads us to a connection between positivity and purity for general d-state systems. Contents

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“…The invariants I 2 , I 3 , and I 4 not only commute with every element of this set, but can also be used to form a representation of the Lie algebra of SU (2) [20]. It has been shown that the encoded, or logical analogues of the Pauli matrices acting on an encoded qubit can be given in terms of these invariants by the relations…”

Section: Three Quditsmentioning

confidence: 99%

“…In addition, the invariant I (α,β) 2 can also be used to perform the generalized exchange interaction between the states |p (α) |q (β) associated with particles α and β since it has been shown in Ref. [20] …”

Section: Three Quditsmentioning

confidence: 99%

Casimir invariants for systems undergoing collective motion

Bishop

Byrd

2011

Phys. Rev. A

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Dicke states are states of a collection of particles which have been under active investigation for several reasons. One reason is that the decay rates of these states can be quite different from a set of independently evolving particles. Another reason is that a particular class of these states are decoherence-free or noiseless with respect to a set of errors. These noiseless states, or more generally subsystems, can avoid certain types of errors in quantum information processing devices. Here we provide a method for calculating invariants of systems of particles undergoing collective motions. These invariants can be used to determine a complete set of commuting observables for a class of Dicke states as well as identify possible logical operations for decoherence-free/noiseless subsystems. Our method is quite general and provides results for cases where the constituent particles have more than two internal states.

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Compatible transformations for a qudit decoherence-free/noiseless encoding

Cited by 13 publications

References 65 publications

Implementation of a simple operator-quantum-error-correction scheme

Implementation of a simple operator-quantum-error-correction scheme

General open-system quantum evolution in terms of affine maps of the polarization vector

Casimir invariants for systems undergoing collective motion

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