arXiv:2301.04989 · Quantum Information and Computation

Qudit Dicke state preparation

Rafael I. Nepomechie, David Raveh

Published Feb 1, 2024• 12 citations

Abstract

Qudit Dicke states are higher-dimensional analogues of an important class of highly-entangled completely symmetric quantum states known as (qubit) Dicke states. A circuit for preparing arbitrary qudit Dicke states deterministically is formulated. An explicit decomposition of the circuit in terms of elementary gates is presented, and is implemented in cirq for the qubit and qutrit cases.

DOI: 10.26421/qic24.1-2-2

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arXiv:2301.04989 · Quantum Information and Computation

Qudit Dicke state preparation

Rafael I. Nepomechie, David Raveh

Published Feb 1, 2024• 12 citations

arXiv HTML PDF

Abstract

DOI: 10.26421/qic24.1-2-2

Related papers

q-analog qudit Dicke states

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Abstract Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their d -level generalization. We define here q -deformed qudit Dicke states using the quantum algebra s u q ( d ) . We show that these states can be compactly expressed as a weighted sum over permutations with q -factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a q -dependence does not change the circuit gate count.Abstract Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their d -level generalization. We define here q -deformed qudit Dicke states using the quantum algebra s u q ( d ) . We show that these states can be compactly expressed as a weighted sum over permutations with q -factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a q -dependence does not change the circuit gate count.

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Simple Ways of Preparing Qudit Dicke States

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Abstract Dicke states are permutation-invariant superpositions of qubit computational basis states, which play a prominent role in quantum information science. We consider here two higher-dimensional generalizations of these states: SU ( d ) spin- s Dicke states and SU ( 2 ) Dicke states. We present various ways of preparing both types of qudit Dicke states on a qudit quantum computer, using two main approaches: a deterministic approach, based on exact canonical matrix product state representations; and a probabilistic approach, based on quantum phase estimation. The quantum circuits are explicit and straightforward, and are arguably simpler than those previously reported.Abstract Dicke states are permutation-invariant superpositions of qubit computational basis states, which play a prominent role in quantum information science. We consider here two higher-dimensional generalizations of these states: SU ( d ) spin- s Dicke states and SU ( 2 ) Dicke states. We present various ways of preparing both types of qudit Dicke states on a qudit quantum computer, using two main approaches: a deterministic approach, based on exact canonical matrix product state representations; and a probabilistic approach, based on quantum phase estimation. The quantum circuits are explicit and straightforward, and are arguably simpler than those previously reported.

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We present short-depth quantum circuits to deterministically prepare any Dicke state $\left| {D_k^n} \right\rangle $, which is the equal-amplitude superposition of all n-qubit computational basis states with Hamming Weight k. Dicke states are an important class of entangled quantum states with a large variety of applications, and a long history of experimental creation in physical systems. On the other hand, not much is known regarding efficient scalable quantum circuits for Dicke state preparation on realistic quantum computing hardware connectivities.Concretely, we present preparation circuits for Dicke states $\left| {D_k^n} \right\rangle $ with (i) depth of $\mathcal{O}\left( {k\log \frac{n}{k}} \right)$ for all-to-all connectivity (such as on current ion trap devices); (ii) depth of $\mathcal{O}\left( {k\surd \frac{n}{k}} \right) = \mathcal{O}(\sqrt {nk} )$ for Grid connectivity on grids of size $\Omega \left( {\surd \frac{n}{s}} \right) \times \mathcal{O}(\sqrt {ns} )$ with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of $\mathcal{O}(kn)$ , need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth $\mathcal{O}(n)$ on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth $\mathcal{O}(\log n)$ on all-to-all connectivity for k = 1 (AQT 2019 [9]).We present short-depth quantum circuits to deterministically prepare any Dicke state $\left| {D_k^n} \right\rangle $, which is the equal-amplitude superposition of all n-qubit computational basis states with Hamming Weight k. Dicke states are an important class of entangled quantum states with a large variety of applications, and a long history of experimental creation in physical systems. On the other hand, not much is known regarding efficient scalable quantum circuits for Dicke state preparation on realistic quantum computing hardware connectivities.Concretely, we present preparation circuits for Dicke states $\left| {D_k^n} \right\rangle $ with (i) depth of $\mathcal{O}\left( {k\log \frac{n}{k}} \right)$ for all-to-all connectivity (such as on current ion trap devices); (ii) depth of $\mathcal{O}\left( {k\surd \frac{n}{k}} \right) = \mathcal{O}(\sqrt {nk} )$ for Grid connectivity on grids of size $\Omega \left( {\surd \frac{n}{s}} \right) \times \mathcal{O}(\sqrt {ns} )$ with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of $\mathcal{O}(kn)$, need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth $\mathcal{O}(n)$ on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth $\mathcal{O}(\log n)$ on all-to-all connectivity for k = 1 (AQT 2019 [9]).

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We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a $1|2$ bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.We present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a $1|2$ bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.

quant-pharXiv:2603.00845Physical Review AMar 3, 20261 citation

(\left|D^n_k \right>)

are an important class of entangled states with uses in many branches of Quantum Information. In this regard we provide the most efficient Deterministic Dicke State Preparation Circuit in terms of CNOT and single qubit gate counts in comparison to existing literature. We further observe that our improvements also reduce architectural constraints of the circuits. We implement the circuit for preparing

\left| D^4_2 \right>

on the "ibmqx2" machine of the IBM QX service and observe that the error induced due to noise in the system is lesser in comparison to the existing circuit descriptions. We conclude by describing the CNOT map of the generic

\left| D^n_k \right>

preparation circuit and analyze different ways of distributing the CNOT gates in the circuit and its affect on the induced error.

\mathcal{O}\left( {k\log \frac{n}{k}} \right)

for all-to-all connectivity (such as on current ion trap devices); (ii) depth of

\mathcal{O}\left( {k\surd \frac{n}{k}} \right) = \mathcal{O}(\sqrt {nk} )

for Grid connectivity on grids of size

\Omega \left( {\surd \frac{n}{s}} \right) \times \mathcal{O}(\sqrt {ns} )

with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of

\mathcal{O}(kn)

, need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth

\mathcal{O}(n)

on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth

\mathcal{O}(\log n)

on all-to-all connectivity for k = 1 (AQT 2019 [9]).

1|2

bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.