# Generation of Pseudo-Random Quantum States on Actual Quantum Processors

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## Abstract

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## 1. Introduction

## 2. Generation of Pseudo-Random Quantum States

#### 2.1. Cartan’s KAK Decomposition of the Unitary Group

#### 2.2. Direct Generation of Two-Qubit Random Quantum States

#### 2.3. Comparison of KAK and Direct Method

## 3. Results on Actual Quantum Hardware

#### 3.1. Comparison between Hardware Platforms

#### 3.2. Entanglement Evolution

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Random State Purities Moments

## Appendix B. Cartan’s KAK Decomposition of the Unitary Group

**Figure A1.**A quantum circuit implementing a two-qubit unitary gate using the KAK parametrization of $SU\left(4\right)$.

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**Figure 1.**The pseudo-random state generator circuit consists of m layers of random permutations of the qubit labels, followed by random two-qubit gates. When the circuit width n is odd, one of the qubits is idle in each layer. In this figure, a circuit with $n=6$ qubits width is shown for illustration purposes.

**Figure 2.**A circuit for two-qubit random state generation. Rotations ${R}_{k}$ are obtained by exponentiating the corresponding Pauli matrices ${\sigma}_{k}$.

**Figure 3.**The pseudo-random state generator circuit consists of m layers of random permutations of the qubit labels, followed by random D gates. In this figure, a circuit with $n=6$ qubit’s width is shown for illustrative purposes.

**Figure 4.**Average mean value relative error (

**left**) and average variance relative error (

**right**) for purities as a function of the number of steps (i.e., layers in the quantum circuit) and the ensemble size for 4-qubit (

**top**), 6-qubit (

**middle**), and 8-qubit (

**bottom**) pseudo-random quantum state. The solid lines represent the direct method while the dashed lines represent the KAK method.

**Figure 5.**Purity mean value (

**top**) and variance (

**bottom**) of a pseudo-random quantum state plotted as a function of partition size. The various colors represent systems of different dimensions (number of qubits). The black dots are the expected values for a true random state. Here is shown the direct method with 20 steps and ${N}_{e}=100$.

**Figure 6.**Architectures of the quantum processors used in this work. The circles represent the qubits while the lines represent the physical connection between them. On the left, we have the architecture of IonQ’s $Harmony$, which clearly shows the complete connectivity of ion-based devices. On the right, we have $ibm\_lagos$. Here, the color scheme (blue for min, violet for max) refers to the single-qubit (color of the circles) and two-qubit (color of the lines) error rates. These are purely indicative since the rates change upon every calibration of the device.

**Figure 7.**Comparison between the purities of a 4- and 6-qubit pseudo-random quantum states, generated in the two different realizations of a quantum computer investigated, with the direct method. In green, the superconductor IBM’s ibm_lagos is shown, while IonQ’s Harmony is shown in blue. Red curves give the results for ideal random states. Data were obtained on 10 September 2022, for ibm_lagos and on 24 July 2022, for Harmony.

**Figure 8.**Evolution of the entanglement content of a pseudo-random quantum state generated by the circuit described in Figure 3 as a function of the number of layers (steps). The panels on the right show the individual curves, with the horizontal solid lines highlighting the purity expectation values for a true random state. The horizontal dashed lines refer to the purity of a maximally mixed state. Data taken from $ibm\_lagos$ on 29 January 2023.

**Table 1.**Table of quantum processing units (QPUs) evaluated in [30] using the quantum volume (QV) protocol. Values of QV, as well as single-qubit (1Q) gate, two-qubit (2Q) gate and state preparation and measurement (SPAM) fidelities are all vendor-provided metrics. The mean gate and SPAM fidelities are computed in [30] across all operations of the same type available on the device during the whole QV circuit execution duration. The number of edges for each backend was simply counted as the number of connections between qubits.

QPU | Fidelity | |||||||
---|---|---|---|---|---|---|---|---|

Vendor | Backend | QV | # Qubit | Topology | # Edges | 2Q Gate | 1Q Gate | SPAM |

IBM Q | $ibm\_lagos$ | 32 | 7 | Falcon r5.11H | 6 | 0.9924 | 0.9998 | 0.9862 |

IonQ | $Harmony$ | 8 * | 11 | All-to-All | 55 | 0.96541 | 0.9972 | 0.99709 |

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**MDPI and ACS Style**

Cenedese, G.; Bondani, M.; Rosa, D.; Benenti, G.
Generation of Pseudo-Random Quantum States on Actual Quantum Processors. *Entropy* **2023**, *25*, 607.
https://doi.org/10.3390/e25040607

**AMA Style**

Cenedese G, Bondani M, Rosa D, Benenti G.
Generation of Pseudo-Random Quantum States on Actual Quantum Processors. *Entropy*. 2023; 25(4):607.
https://doi.org/10.3390/e25040607

**Chicago/Turabian Style**

Cenedese, Gabriele, Maria Bondani, Dario Rosa, and Giuliano Benenti.
2023. "Generation of Pseudo-Random Quantum States on Actual Quantum Processors" *Entropy* 25, no. 4: 607.
https://doi.org/10.3390/e25040607