arXiv:2306.13146 · Physical Review A

Entropy cones and entanglement evolution for Dicke states

William Munizzi, Howard J. Schnitzer

Published Jan 3, 2024• 8 citations

Abstract

The $N$-qubit Dicke states $|{D}_{k}^{N}\ensuremath{\rangle}$, of Hamming weight $k$, are a class of entangled states which play an important role in quantum algorithm optimization. We present a general calculation of entanglement entropy in Dicke states, which we use to describe the $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy cone. We demonstrate that all $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy vectors emerge symmetrized and use this to define a min-cut protocol on star graphs which realizes $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy vectors. We identify the stabilizer group for all $|{D}_{k}^{N}\ensuremath{\rangle}$, under the action of the $N$-qubit Pauli group and two-qubit Clifford group, which we use to construct $|{D}_{k}^{N}\ensuremath{\rangle}$ reachability graphs. We use these reachability graphs to analyze and bound evolution of $|{D}_{k}^{N}\ensuremath{\rangle}$ entropy vectors in Clifford circuits.

DOI: 10.1103/physreva.109.012405

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arXiv:2306.13146 · Physical Review A

Entropy cones and entanglement evolution for Dicke states

William Munizzi, Howard J. Schnitzer

Published Jan 3, 2024• 8 citations

arXiv HTML PDF

Abstract

The

N

-qubit Dicke states

|{D}_{k}^{N}\ensuremath{\rangle}

, of Hamming weight

k

, are a class of entangled states which play an important role in quantum algorithm optimization. We present a general calculation of entanglement entropy in Dicke states, which we use to describe the

|{D}_{k}^{N}\ensuremath{\rangle}

entropy cone. We demonstrate that all

|{D}_{k}^{N}\ensuremath{\rangle}

entropy vectors emerge symmetrized and use this to define a min-cut protocol on star graphs which realizes

|{D}_{k}^{N}\ensuremath{\rangle}

entropy vectors. We identify the stabilizer group for all

|{D}_{k}^{N}\ensuremath{\rangle}

, under the action of the

N

-qubit Pauli group and two-qubit Clifford group, which we use to construct

|{D}_{k}^{N}\ensuremath{\rangle}

reachability graphs. We use these reachability graphs to analyze and bound evolution of

|{D}_{k}^{N}\ensuremath{\rangle}

entropy vectors in Clifford circuits.

DOI: 10.1103/physreva.109.012405

Related papers

Bounding entanglement entropy with Clifford double cosets

Cynthia A Keeler, William Munizzi, Jason Pollack

Following on our previous work arXiv:2204.07593 and arXiv:2306.01043 studying the orbits of quantum states under Clifford circuits via `reachability graphs', we introduce `contracted graphs' whose vertices represent classes of quantum states with the same entropy vector. These contracted graphs represent the double cosets of the Clifford group, where the left cosets are built from the stabilizer subgroup of the starting state and the right cosets are built from the entropy-preserving operators. We study contracted graphs for stabilizer states, as well as W states and Dicke states, discussing how the diameter of a state's contracted graph constrains the `entropic diversity' of its $2$ -qubit Clifford orbit. We derive an upper bound on the number of entropy vectors that can be generated using any $n$ -qubit Clifford circuit, for any quantum state. We speculate on the holographic implications for the relative proximity of gravitational duals of states within the same Clifford orbit. Although we concentrate on how entropy evolves under the Clifford group, our double-coset formalism, and thus the contracted graph picture, is extendable to generic gate sets and generic state properties.Following on our previous work arXiv:2204.07593 and arXiv:2306.01043 studying the orbits of quantum states under Clifford circuits via `reachability graphs', we introduce `contracted graphs' whose vertices represent classes of quantum states with the same entropy vector. These contracted graphs represent the double cosets of the Clifford group, where the left cosets are built from the stabilizer subgroup of the starting state and the right cosets are built from the entropy-preserving operators. We study contracted graphs for stabilizer states, as well as W states and Dicke states, discussing how the diameter of a state's contracted graph constrains the `entropic diversity' of its $2$-qubit Clifford orbit. We derive an upper bound on the number of entropy vectors that can be generated using any $n$-qubit Clifford circuit, for any quantum state. We speculate on the holographic implications for the relative proximity of gravitational duals of states within the same Clifford orbit. Although we concentrate on how entropy evolves under the Clifford group, our double-coset formalism, and thus the contracted graph picture, is extendable to generic gate sets and generic state properties.

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We introduce a method for studying state preparation complexity in dense quantum $p$ -spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$ . At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $Ω(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$ . In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+ $T$ circuits with $o(n)$ $T$ -gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $Ω(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

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Short-Depth Circuits for Dicke State Preparation

Andreas Bärtschi, Stephan Eidenbenz

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]).

arXiv:2207.09998IEEE International Conference on Quantum Computing and Engineering (QCE)Sep 1, 202243 citations

Entropic lens on stabilizer states

Cynthia A Keeler, William Munizzi, Jason Pollack

The $n$ -qubit stabilizer states are those left invariant by a ${2}^{n}$ -element subset of the Pauli group. The Clifford group is the group of unitaries which take stabilizer states to stabilizer states; a physically motivated generating set, the Hadamard, phase, and controlled-not (cnot) gates which comprise the Clifford gates, impose a graph structure on the set of stabilizers. We explicitly construct these structures, the ``reachability graphs,'' at $n\ensuremath{\le}5$. When we consider only a subset of the Clifford gates, the reachability graphs separate into multiple, often complicated, connected components. Seeking an understanding of the entropic structure of the stabilizer states, which is ultimately built up by cnot gate applications on two qubits, we are motivated to consider the restricted subgraphs built from the Hadamard and cnot gates acting on only two of the $n$ qubits. We show how the two subgraphs already present at two qubits are embedded into more complicated subgraphs at three and four qubits. We argue that no additional types of subgraph appear beyond four qubits, but that the entropic structures within the subgraphs can grow progressively more complicated as the qubit number increases. Starting at four qubits, some of the stabilizer states have entropy vectors which are not allowed by holographic entropy inequalities. We comment on the nature of the transition between holographic and nonholographic states within the stabilizer reachability graphs.The $n$-qubit stabilizer states are those left invariant by a ${2}^{n}$-element subset of the Pauli group. The Clifford group is the group of unitaries which take stabilizer states to stabilizer states; a physically motivated generating set, the Hadamard, phase, and controlled-not (cnot) gates which comprise the Clifford gates, impose a graph structure on the set of stabilizers. We explicitly construct these structures, the ``reachability graphs,'' at $n\ensuremath{\le}5$. When we consider only a subset of the Clifford gates, the reachability graphs separate into multiple, often complicated, connected components. Seeking an understanding of the entropic structure of the stabilizer states, which is ultimately built up by cnot gate applications on two qubits, we are motivated to consider the restricted subgraphs built from the Hadamard and cnot gates acting on only two of the $n$ qubits. We show how the two subgraphs already present at two qubits are embedded into more complicated subgraphs at three and four qubits. We argue that no additional types of subgraph appear beyond four qubits, but that the entropic structures within the subgraphs can grow progressively more complicated as the qubit number increases. Starting at four qubits, some of the stabilizer states have entropy vectors which are not allowed by holographic entropy inequalities. We comment on the nature of the transition between holographic and nonholographic states within the stabilizer reachability graphs.

arXiv:2204.07593Physical Review ADec 16, 20228 citations

The

N

-qubit Dicke states

|{D}_{k}^{N}\ensuremath{\rangle}

, of Hamming weight

k

|{D}_{k}^{N}\ensuremath{\rangle}

entropy cone. We demonstrate that all

|{D}_{k}^{N}\ensuremath{\rangle}

entropy vectors emerge symmetrized and use this to define a min-cut protocol on star graphs which realizes

|{D}_{k}^{N}\ensuremath{\rangle}

entropy vectors. We identify the stabilizer group for all

|{D}_{k}^{N}\ensuremath{\rangle}

, under the action of the

N

-qubit Pauli group and two-qubit Clifford group, which we use to construct

|{D}_{k}^{N}\ensuremath{\rangle}

reachability graphs. We use these reachability graphs to analyze and bound evolution of

|{D}_{k}^{N}\ensuremath{\rangle}

entropy vectors in Clifford circuits.

2

-qubit Clifford orbit. We derive an upper bound on the number of entropy vectors that can be generated using any

n

-qubit Clifford circuit, for any quantum state. We speculate on the holographic implications for the relative proximity of gravitational duals of states within the same Clifford orbit. Although we concentrate on how entropy evolves under the Clifford group, our double-coset formalism, and thus the contracted graph picture, is extendable to generic gate sets and generic state properties.

We introduce a method for studying state preparation complexity in dense quantum

p

-spin Hamiltonians on

n

qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly

p

qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of

\sqrt n

for sufficiently large fixed

p

. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires

Ω(n^2/\log n)

one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width

O(n)

. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+

T

circuits with

o(n)

T

-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

The

n

-qubit

k

-weight Dicke states

|D^n_k\rangle

, defined as the uniform superposition of all computational basis states with exactly

k

qubits in state

|1\rangle

, form a basis of the symmetric subspace and represent an important class of entangled quantum states with broad applications in quantum computing. We propose deterministic quantum circuits for Dicke state preparation under two commonly seen qubit connectivity constraints: 1. All-to-all qubit connectivity: our circuit has depth

O(\log(k)\log(n/k)+k)

, which improves the previous best bound of

O(k\log(n/k))

. 2. Grid qubit connectivity (

(n_1\times n_2)

-grid,

n_1\le n_2

): (a) For

k\ge n_2/n_1

, we design a circuit with depth

O(k\log(n/k)+n_2)

, surpassing the prior

O(\sqrt{nk})

bound. (b) For

k< n_2/n_1

, we design an optimal-depth circuit with depth

O(n_2)

. Furthermore, we establish the depth lower bounds of

Ω(\log(n))

for all-to-all qubit connectivity and

Ω(n_2)

for

(n_1\times n_2)

-grid connectivity constraints, demonstrating the near-optimality of our constructions.

\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]).

The

n

-qubit stabilizer states are those left invariant by a

{2}^{n}

-element subset of the Pauli group. The Clifford group is the group of unitaries which take stabilizer states to stabilizer states; a physically motivated generating set, the Hadamard, phase, and controlled-not (cnot) gates which comprise the Clifford gates, impose a graph structure on the set of stabilizers. We explicitly construct these structures, the ``reachability graphs,'' at $n\ensuremath{\le}5$. When we consider only a subset of the Clifford gates, the reachability graphs separate into multiple, often complicated, connected components. Seeking an understanding of the entropic structure of the stabilizer states, which is ultimately built up by cnot gate applications on two qubits, we are motivated to consider the restricted subgraphs built from the Hadamard and cnot gates acting on only two of the

n

qubits. We show how the two subgraphs already present at two qubits are embedded into more complicated subgraphs at three and four qubits. We argue that no additional types of subgraph appear beyond four qubits, but that the entropic structures within the subgraphs can grow progressively more complicated as the qubit number increases. Starting at four qubits, some of the stabilizer states have entropy vectors which are not allowed by holographic entropy inequalities. We comment on the nature of the transition between holographic and nonholographic states within the stabilizer reachability graphs.