ORIGINAL RESEARCH article
Sec. Quantum Engineering
Volume 1 - 2022 | https://doi.org/10.3389/frqst.2022.1026025
Experimental validation of the Kibble-Zurek mechanism on a digital quantum computer
- 1Departamento de Física, Universidad de los Andes, Bogotá, Colombia
- 2Instituto de Física Fundamental IFF-CSIC, Madrid, Spain
The Kibble-Zurek mechanism (KZM) captures the essential physics of nonequilibrium quantum phase transitions with symmetry breaking. KZM predicts a universal scaling power law for the defect density which is fully determined by the system’s critical exponents at equilibrium and the quenching rate. We experimentally tested the KZM for the simplest quantum case, a single qubit under the Landau-Zener evolution, on an open access IBM quantum computer (IBM-Q). We find that for this simple one-qubit model, experimental data validates the central KZM assumption of the adiabatic-impulse approximation for a well isolated qubit. Furthermore, we report on extensive IBM-Q experiments on individual qubits embedded in different circuit environments and topologies, separately elucidating the role of crosstalk between qubits and the increasing decoherence effects associated with the quantum circuit depth on the KZM predictions. Our results strongly suggest that increasing circuit depth acts as a decoherence source, producing a rapid deviation of experimental data from theoretical unitary predictions.
Characterizing the non-equilibrium dynamics in noisy intermediate-scale quantum (NISQ) devices plays an important role in developing both hardware and architecture designs in the search for scalable quantum computers. NISQ devices have recently attracted tremendous interest, resulting in rapid progress in fundamental studies of novel hardware and architecture together with promising potential for quantum computing (Preskill, 2018; Bharti et al., 2022). For example, advancements in NISQ devices demonstrate a “quantum advantage” in solving sampling problems (Arute et al., 2019; Zhong et al., 2020; Mooney et al., 2021). To further improve quantum advantage, it is desirable that devices show important features such as high-fidelity gates, qubits with long coherence times, control of state preparation and measurement. (Flammia and Liu, 2011; da Silva et al., 2011; Proctor et al., 2019). Open-access/online NISQ devices have recently become readily available, such as those provided publicly by the IBM Quantum Experience platform (IBM-Corporation, 2022), showing a significant improvement in the last few years. Despite suffering from noise and scalability limitations, this platform offers a unique possibility to experiment with actual few qubit quantum devices in order to carry out a rigorous study of dynamical quantum properties in different settings along the real time-dynamics of quantum hardware. A key feature of merit in the current NISQ regime is the ability to simulate non-equilibrium quantum dynamics. The Kibble-Zurek mechanism (KZM) (Kibble, 1976; Kibble, 1980; Zurek, 1985; Zurek, 1993) is a prominent paradigm to unravel signatures of universal dynamics in the scenario of a finite-rate spontaneous symmetry breaking. The KZM predicts the production of topological defects (kinks, vortices, strings) or in general, non-equilibrium excitations (in both short- and large-ranged interacting systems) in the course of either quantum (Dziarmaga, 2005; Zurek et al., 2005; Acevedo et al., 2014) or classical (Kibble, 1980; Zurek, 1985) phase transitions. The key result of KZM is concerned with the fact that the mean value of density of topological defects scales as a power law of the quench rate. Furthermore, new evidence of scaling in the high-order cumulants has also been recently shown (Del Campo, 2018; Gómez-Ruiz et al., 2020). These theoretical predictions have been observed in various experimental platforms such as Bose Gas (Goo et al., 2021), trapped ions (Cui et al., 2020), quantum annealer (Bando et al., 2020; King et al., 2022), Bose-Einstein Condensate (Damski and Zurek, 2007; Anquez et al., 2016) and Rydberg atoms (Keesling et al., 2019).
Damski et al. (Damski, 2005; Damski and Zurek, 2006; Cucchietti et al., 2007) established a close relationship between second order quantum phase transitions and avoided level crossing evolutions, thus establishing the Landau-Zener (LZ) model itself as the simplest paradigmatic scenario for probing KZM (Landau, 1932a; Landau, 1932b; Majorana, 1932; Stückelberg, 1932; Zener and Fowler, 1932). The density of topological defects can be expressed as a transition probability for a two-level system. Therefore, this relationship can be tested in generic single qubit platforms. This relationship has been probed by using optical interferometry (Xu et al., 2014), superconducting qubits (Wang et al., 2014; Gong et al., 2016) and trapped ion systems (Cui et al., 2016).
IBM-Q currently grants access up to 5-qubit quantum machines based on superconducting transmon qubits which are controllable using Qiskit, an open-source software development kit (Aleksandrowicz et al., 2019; Andersson et al., 2020). These machines have been successfully utilized in simulating spin models (Cervera-Lierta, 2018; Rodriguez-Vega et al., 2022), topological fermionic models (Koh et al., 2022), quantum entanglement (Choo et al., 2018; Wang et al., 2018; Cruz et al., 2019; Mooney et al., 2019; Pozzobom and Maziero, 2019), far-from-equilibrium dynamics (Zhukov et al., 2018), non-equilibrium quantum thermodynamics (Gherardini et al., 2021; Solfanelli et al., 2021), open-quantum systems (García-Pérez et al., 2020), among others. One of the future advantages of IBM-Q is the possibility to do simulation of quantum systems beyond the maximum limits of classical computer over a wide range of parameters. In this work, we test the KZM adiabatic-impulse assumption on the simplest, but important case of a single qubit (LZ model), through experiments on the Qiskit (Andersson et al., 2020) simulator and real quantum hardware, establishing the limits required to obtain accurate results in each case. We successfully reproduced the LZ dynamics under a discrete time evolution in current IBM quantum devices which can provide information about dynamics state evolution given that error mitigation procedures were implemented. Additionally, noticeable effects of decoherence are observed and explained by a simple phenomenological model of relaxation and dephasing for open quantum systems. Furthermore, analysis and estimation of the experimental asymptotic probability allows us to verify the universal KZM in a timescale appropriate for an almost closed system under an adiabatic quench regime. In summary, the key achievement of this work has been the validation of a central premise of KZM through a protocol to characterize and obtain an effective time-dependent dynamics on IBM realistic quantum computers. For reaching such goal we performed LZ evolution under different annealing times, maintaining a fixed number of total gates, a basic benchmark procedure on quantum critical phenomena in near term quantum computers.
This paper is organized as follows. A brief review on KZM, the LZ model and its close connection with KZM are presented in Sect. 1. In Sect. 2 we present the experimental platform. The contrast between theoretical predictions and experimental results is collected in Sect. 3. Finally, we summarize the main conclusions in Sect. 4.
1 Theoretical background
1.1 Brief review of the Kibble-Zurek Mechanism
The KZM describes the dynamics of a system across a continuous symmetry breaking second-order phase transition induced by the change of a control parameter λ. When the system is driven through the critical point λc, both the correlation length ξ and reaction time τ diverge as
where D is the dimensionality of the system. This result was initially derived in the classical domain (Kibble, 1980; Zurek, 1985) and subsequently extended to quantum systems (Dziarmaga, 2005; Zurek et al., 2005). Additionally, the KZM has also been extended to novel scenarios including long-range interactions (Acevedo et al., 2014; Puebla et al., 2019; Puebla et al., 2020), inhomogeneous systems (Collura and Karevski, 2010; Dziarmaga and Rams, 2010; Gómez-Ruiz and del Campo, 2019) and nonlinear quenches (Barankov and Polkovnikov, 2008; Sen et al., 2008).
FIGURE 1. Connection between KZM and avoided level crossing in a LZ transition. (A) In a continuous second order phase transition, the reaction time diverges near the critical point. The KZM approximation takes into account the total dynamics divided in three stages (adiabatic, impulse and adiabatic) represented by the graduated yellow-dark red-yellow colors and separated by the freeze out-time
1.2 Landau-Zener model
Consider a two-level system, with gap Δ, described by the time-dependent Hamiltonian
where δ = Δ2ta/4 is the adiabaticity parameter. Moreover, χ1 and χ2 are found from the initial condition at z = zi (see the section: Supplementary Data for details of the calculations and derivations):
Notice that, Eqs 4, 5 are valid for any arbitrary initial condition and final time t. For the experimental implementation discussed below, we are interested in studying the system’s evolution from an initial state starting in the anticrossing point at t = 0. In the section: Supplementary Data, the formal solutions for this particular initial condition are summarized.
1.3 Connection between the KZM and LZ evolution
Here we demonstrate how we can implement a controllable evolution using an IBM-Q quantum simulation, in close analogy to the topological defect formation in KZM. Following the seminal arguments exposed in Ref. (Damski, 2005; Damski and Zurek, 2006), topological defects can be built into the LZ model by being associated to the diabatic states. Consider one of the states, such as
Then, a system evolving in time under the LZ model can be used to study transitions between the phases through the probabilities of the diabatic states. The similarity between the reaction time of a second order phase transition and the inverse of energy gap in the LZ Hamiltonian is shown in Figure 1B. In analogy with the KZM, this suggests that the adiabatic-impulse-adiabatic approximation (AI) may be used to estimate the asymptotic probability when the system traverses the avoided level crossing, thus elucidating the link between the KZM and LZ evolution.
We divided the dynamics through the anti-crossing into three stages like the AI scenario for KZM. Without loss of generality, we assume that the system starts at ti → −∞ from the ground state
The AI assumes that the evolution wave function
• Adiabatic dynamics: from ti = −∞ to
• Impulse dynamics: from
• Adiabatic dynamics: from
Where Φ1, Φ2 are global phases, and A is a constant. Following the AI, Damski in Refs (Damski, 2005; Damski and Zurek, 2006). reported the probability of finding the LZ system in the excited state at tf ≫ tLZ, a calculation we briefly summarize for the sake of completeness in view of our main experimental in terest.
From now on, we focus on the LZ dynamics for the evolution starting in the ground state at the anticrossing point. The initial state at t = 0 is then expressed as
Where we have fixed the two-level system gap to Δ = 1. Additionally,
which will be relevant for testing the predictions of the universal AI for KZM below.
2 Experimental IBM-Q platform
We implemented our experimental studies in two topologies or processors types. Figure 2A shows the device layout for the IBMQ 5-qubit ibmq_bogota (Falcon r5.11L topology
FIGURE 2. Decoherence times in different IBM-Q and IBM-Q circuit simulation of the Landau-Zener process. (A) In the pie-like chart, we contrast the thermal relaxation time (T1) and dephasing time (T2), in
3.1 Simulation of the Landau-Zener evolution on IBM-Q
Unitary dynamics.– We are interested in the experimental determination, and respective simulation, on a digital open-access IBM-Q of a single qubit evolution under a linearly time-dependent Hamiltonian (LZ problem). At time ti, a qubit in the processor is initialized in the state
Since we are interested in the evolution from an initial condition where the LZ system is prepared in an equal weight superposition at the anticrossing point, we start by applying the unitary rotation
IBM-Q devices are equipped with the finite and complete set
As already stated, the Landau-Zener dynamics can be exactly solved (see Supplementary Data), thus allowing a direct benchmark test of the experimental results on a realistic quantum device against exact results. For a LZ evolution starting at the anticrossing ground state, we obtain the LZ transition probability PLZ(t) at time t given as
with the amplitudes χ1 and χ2, see Eq. 5, given by:
Where z and δ are given in Section 1.2.
Our first aim is to benchmark our LZ experimental results with the above exact theoretical prediction. This is schematically illustrated in Figure 2C where we display the exact result, see Eq. 13, and a hypothetical grid of points representing expected target data with a separation dt = (tf − ti)/Nt, being Nt the total circuit depth. For every experimental data, 5,000 shots have been realized on each quantum circuit,
FIGURE 3. Measurement of LZ probabilities on IBM-Q. In panel (A), we establish a contrast between the exact and experimental results for the LZ transition probability as a function the number of layers or circuit depth N, and the annealing time ta. In this panel, all figures share the same color vertical scale. The initial condition,
In the next subsection, we address the influence of the number of layers in the LZ simulation circuit and the role of decoherence.
Open system dynamics.- The performance of the hardware worsens with an increasing depth of the circuit. The assumption of a closed quantum system rapidly breaks down for qubits with short relaxation (T1) and dephasing (T2) timescales, thus requiring for a theoretical analysis that resorts to a quantum open system approach. The effects of quantum decoherence are noticeable in the measured probability when scaling the number of gates due to the increase in computing times. We model every qubit on IBM-Q as a two-level system coupled to a Markovian bath. The system evolution is described by a continuous map
In Figure 4, we establish a contrast between the unitary exact dynamics, numerical Lindblad dynamics (QuTip) and the experimental results obtained for qubit 4, the noisiest qubit in both quantum machines. QuTiP is an open-source framework for Python that allows for numerical simulations of quantum dynamics of open systems under different solvers (Johansson et al., 2012; Johansson et al., 2013). Specifically, we depicted the Landau-Zener probability as a function of the number of layers in the circuit, N, for two specific annealing times ta = 1 (colors green/purple) and ta = 0.1 (colors blue/orange). Additionally, we show as an inset the ratio between T2/T1, the bar scale shows the value of this proportion from 0 to 2. Although, ibmq_lima quantum computer has the ratio T2/T1 almost constant, qubit 4 is the most prone to decoherence.
FIGURE 4. Contrast between close and open quantum dynamics for LZ on IBM-Q. The Landau-Zener transition probability is shown as a function of the number of layers N in the circuit implemention for qubit 4, the noisiest qubit for each
3.2 Simulation of the Kibble-Zurek mechanism on IBM-Q
The main purpose of this work is to validate the adiabatic-impulse approximation of the Kibble-Zurek mechanism through the nonequilibrium dynamics of the Landau-Zener model on IBM-Q. Using Eq. 13 with Δ = 1, the asymptotic probability can be exactly calculated as
Expanding the asymptotic probability into series, we obtain (Damski, 2005; Damski and Zurek, 2006)
We find the value of η by directly comparing the adiabatic-impulse approximation given by Eq. 10 and the expansion of the LZ asymptotic probability at first-order (η = π/4). However, non-trivial corrections for high-order terms appear. In both main panels of Figures 5A,B, we depict the agreement of the theoretical prediction for the adiabatic-impulse approximation (Eq. 9) and asymptotic Landau-Zener probability (Eq. 16). We note the role of the corrections for large quench times. For finite-time LZ simulations, estimating the asymptotic transition probability becomes challenging and similar to experimental data. To this end, we introduced the Landau-Zener jump-time t⋆ as the fist zero in the second derivative of the Landau-Zener probability, thus:
FIGURE 5. Simulation of the Kibble-Zurek mechanism on IBM-Q. In both upper and lower panels, we contrast the adiabatic-impulse approximation (Eq. 9), asymptotic Landau-Zener probability (Eq. 16), and the experimental data. In panel (A), we show experimental data retrieved from ibmq_bogota. In panel (B), we present the experimental results from ibmq_lima. In the inset, we present the protocol to calculate the asymptotic experimental Landau-Zener probability. The error bars with length 2ζEB, calculated from the finite-time effect, are also shown. Solid symbols are consistent with the best qubit behavior as depicted in Figure 3.
In this way, we propose that the estimated finite-time asymptotic Landau-Zener probability can be approximated by the average of all values of
For the qubit with the largest decoherence T1 and T2 times (the best qubit from now on), the experimental data show an excellent agreement with the theoretical predictions for the impulse-adiabatic approximation. For large annealing time ta, the experimental data has a significant deviation for some qubits in the ibmq_bogota quantum computer. Indeed, the adiabatic-impulse approximation relationship with the Landau-Zener problem assumes a close system’s quantum dynamics. However, since IBM-Q is benchmarked as an open-quantum system, deviations are to be expected.
In order to further testing the KZM adiabatic-impulse approximation, from our experimental data, we rewrite the Eq. 9 in terms of 3 fitting parameters, as
In Figure 6, we depict the comparison of the fitting parameters x1, x2 and x3 for the best qubit at ibmq_bogota and ibmq_lima. The structure of the fitting expression allows us a direct comparison with the theoretical predictions
FIGURE 6. KZM adiabatic-impulse approximation fitting parameters. From the best qubit experimental data (solid symbols in Figure 5), the fitting to the KZM adiabatic-impulse approximation PAI (ta) given by Eq. 19 is probed (the dashed gray lines correspond to the theoretical predictions). The experimental data at ibmq_bogota and ibmq_lima are depicted in blue and red, respectively.
In this work we explored the dynamics of a two level system under the time-dependent Landau-Zener Hamiltonian on digital IBM Quantum computers. Time evolution was simulated by discretization of the time dependent Hamiltonian and application of subsequent single-qubit unitary gates representing finite time progressions. We studied the Landau Zener transition probability as a function of time by running parallel quantum circuits on 5-qubit machines ibmq_lima and ibmq_bogota with different topologies. We find a strong agreement with the theoretical solution of the LZ problem for robust qubits from both machines. We also considered the effect of decoherence on an open LZ system, modeling the dissipation using collapse operators for relaxation and dephasing. For greater trotterizations of the time evolution operator, increasing computing time cause noticeable deviations from the theoretical LZ solution. The numerical solution of the Lindblad master equation accurately depicts the open system’s relaxation towards the ground state, supported by the measured probabilities.
The above positive LZ results allowed us to demonstrate the first simulation on a realistic quantum computer of the universal Kibble-Zurek mechanism by estimating the asymptotic transition probability obtained from LZ experimental data. Results show excellent agreement for the best qubits considered in each device and low annealing times. We find that larger annealing times demand a greater time resolution in the evolution operator discretization, putting practical limits on the performance achieved, as it becomes limited by the conflict between computing depth and decoherence times. However, the rapid rate of quantum hardware advances may soon change this. Furthermore, an interesting follow-up research direction would consist in focusing on richer open quantum platforms, where KZM has been poorly explored. Thus, using real quantum hardware to test quantum universal dynamical behaviors, in both closed and open systems, represent an interesting extension of the results presented in this work.
Data availability statement
The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below: https://github.com/sanhq17/Testing_KZM_IBMQ.
FG-R and LQ initiated and guided the project. SH-Q took the experimental measurements. FG-R developed numerical simulations and prepared the figures. All authors contributed to the analysis of the results and the writing of the manuscript.
SH-Q, FR, and LQ are thankful for the financial support from Facultad de Ciencias-UniAndes projects: INV-2021-128-2292, and INV-2019-84-1841. FG-R acknowledges financial support from European Commission FET-Open project AVaQus GA 899561.
The authors thank to Bogdan Damski for useful comments and suggestions.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frqst.2022.1026025/full#supplementary-material
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Keywords: IBM quantum computing, Kibble-Zurek mechanism, Landau-Zener model, adiabatic-impulse approximation, quantum technologies
Citation: Higuera-Quintero S, Rodríguez FJ, Quiroga L and Gómez-Ruiz FJ (2022) Experimental validation of the Kibble-Zurek mechanism on a digital quantum computer. Front. Quantum. Sci. Technol. 1:1026025. doi: 10.3389/frqst.2022.1026025
Received: 23 August 2022; Accepted: 07 October 2022;
Published: 25 October 2022.
Edited by:Erik Torrontegui, Universidad Carlos III de Madrid, Spain
Reviewed by:Francesco Plastina, University of Calabria, Italy
Ricardo Puebla, Universidad Carlos III de Madrid, Spain
Copyright © 2022 Higuera-Quintero, Rodríguez, Quiroga and Gómez-Ruiz. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Fernando J. Gómez-Ruiz, email@example.com