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arXiv:2007.01681

On Actual Preparation of Dicke State on a Quantum Computer

Chandra Sekhar Mukherjee, Subhamoy Maitra, Vineet Gaurav, Dibyendu Roy

Published Jul 3, 2020• 4 citations
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Abstract

The exact number of CNOT and single qubit gates needed to implement a Quantum Algorithm in a given architecture is one of the central problems of Quantum Computation. In this work we study the importance of concise realizations of Partially defined Unitary Transformations for better circuit construction using the case study of Dicke State Preparation. The Dicke States $(\left|D^n_k \right>)$ 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.

DOI
10.48550/arxiv.2007.01681

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The exact number of CNOT and single qubit gates needed to implement a Quantum Algorithm in a given architecture is one of the central problems of Quantum Computation. In this work we study the importance of concise realizations of Partially defined Unitary Transformations for better circuit construction using the case study of Dicke State Preparation. The Dicke States (∣Dkn>)(\left|D^n_k \right>)(∣Dkn​⟩) 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 ∣D24>\left| D^4_2 \right>​D24​⟩ 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 ∣Dkn>\left| D^n_k \right>∣Dkn​⟩ preparation circuit and analyze different ways of distributing the CNOT gates in the circuit and its affect on the induced error.
Exact requirement of controlled NOT (CNOT) and single-qubit gates to implement a quantum algorithm in a given architecture is one of the central problems in this computational paradigm. In this article, we take a tutorial approach in explaining the preparation of Dicke states (|Dkn〉) using concise realizations of partially defined unitary transformations. We show how to efficiently implement the state-of-the-art deterministic Dicke state preparation circuits and in turn optimize them in terms of CNOT and single-qubit gate counts. We explain theoretical ideas in reducing the gate counts and observe how these improvements are reflected in actual implementation of the circuits. To emphasize the advantages, we describe the circuit for preparing |D24〉 on the “ibmqx2” machine of the IBM quantum experience (QX) service. Our approach shows that the error induced due to noise in the system is lesser in comparison to the existing works. We conclude by describing the CNOT map of the generic |Dkn〉 preparation circuit and analyze different ways of distributing the CNOT gates in the circuit and its effect on the induced error in the Noisy Intermediate Scale Quantum environment.
Minimizing the use of CNOT gates in quantum state preparation is a crucial step in quantum compilation, as they introduce coupling constraints and more noise than single-qubit gates. Reducing the number of CNOT gates can lead to more efficient and accurate quantum computations. However, the attainment of optimal solutions is hindered by the complexity of modeling superposition and entanglement. In this paper, we propose an effective state preparation algorithm using an exact CNOT synthesis formulation. Our method represents a milestone as the first design automation algorithm to surpass manual design, reducing the best CNOT count to prepare a Dicke state by 2x. For general states with up to 20 qubits, our method reduces the CNOT count by 9% and 32% for dense and sparse states, respectively, on average, compared to the latest algorithms.
In quantum computation every unitary operation can be decomposed into quantum circuits---a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not(cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of 222. It also reduces the total number of cnot gates from 2n{2}^{n}2n to 23242n\frac{23}{24}{2}^{n}2423​2n in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from 111111 cnot gates to 999 and the depth from 111111 to 555 steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.
The nnn-qubit kkk-weight Dicke states ∣Dkn⟩|D^n_k\rangle∣Dkn​⟩, defined as the uniform superposition of all computational basis states with exactly kkk qubits in state ∣1⟩|1\rangle∣1⟩, 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)O(\log(k)\log(n/k)+k)O(log(k)log(n/k)+k), which improves the previous best bound of O(klog⁡(n/k))O(k\log(n/k))O(klog(n/k)). 2. Grid qubit connectivity ((n1×n2)(n_1\times n_2)(n1​×n2​)-grid, n1≤n2n_1\le n_2n1​≤n2​): (a) For k≥n2/n1k\ge n_2/n_1k≥n2​/n1​, we design a circuit with depth O(klog⁡(n/k)+n2)O(k\log(n/k)+n_2)O(klog(n/k)+n2​), surpassing the prior O(nk)O(\sqrt{nk})O(nk​) bound. (b) For k<n2/n1k< n_2/n_1k<n2​/n1​, we design an optimal-depth circuit with depth O(n2)O(n_2)O(n2​). Furthermore, we establish the depth lower bounds of Ω(log⁡(n))Ω(\log(n))Ω(log(n)) for all-to-all qubit connectivity and Ω(n2)Ω(n_2)Ω(n2​) for (n1×n2)(n_1\times n_2)(n1​×n2​)-grid connectivity constraints, demonstrating the near-optimality of our constructions.
We perform optimal-control-theory calculations to determine the minimum number of two-qubit controlled-not (cnot) gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that all produce the desired result, even at the minimum number of gates. We also show that the number of entangling gates can be reduced if we use multiqubit entangling gates instead of two-qubit cnot gates, as one might expect based on parameter counting calculations. In addition to treating the general case of arbitrary target states or unitary operators, we apply the numerical approach to the special case of synthesizing the multiqubit Toffoli gate. This approach can be used to investigate any other specific few-qubit task and provides insight into the tightness of different bounds in the literature.
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 O(klog⁡nk)\mathcal{O}\left( {k\log \frac{n}{k}} \right)O(klogkn​) for all-to-all connectivity (such as on current ion trap devices); (ii) depth of O(k√nk)=O(nk)\mathcal{O}\left( {k\surd \frac{n}{k}} \right) = \mathcal{O}(\sqrt {nk} )O(k√kn​)=O(nk​) for Grid connectivity on grids of size Ω(√ns)×O(ns)\Omega \left( {\surd \frac{n}{s}} \right) \times \mathcal{O}(\sqrt {ns} )Ω(√sn​)×O(ns​) with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of O(kn)\mathcal{O}(kn)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 O(n)\mathcal{O}(n)O(n) on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth O(log⁡n)\mathcal{O}(\log n)O(logn) on all-to-all connectivity for k = 1 (AQT 2019 [9]).