# Simulating a Discrete Time Crystal Using Qiskit

*By Caleb Clothier*

**Space and Time Crystals: An Analogy**

You may have recently heard about experimental realizations of time crystals on quantum systems. In this blog post, we will explore some of the physics underlying the time crystal, and see how to implement and simulate our own discrete time crystal in Qiskit using IBM Quantum’s powerful cloud simulators. A companion Jupyter notebook to this article can be found here.

Time crystals were originally conceived in 2012 by Nobel laureate Frank Wilczek as a temporal analog to the more familiar phenomena of “space crystals,” for example, diamonds or quartz. A defining feature of these ordinary crystals is a high degree of spatial organization: the atoms or molecules of a crystalline solid form a *stable*, *periodic* structure in space.

In the language of theoretical physics, a space crystal is a phase of matter that is said to spontaneously break *space-translation symmetry. *The laws of physics describing the evolution of atoms are the same everywhere, invariant under arbitrary movements or rotations through space. Yet the stable, lowest-energy equilibrium state of a system like a crystalline lattice repeating at discrete spatial intervals violates the continuous space-translation symmetry that is respected by the underlying physical laws. Whereas the equations governing the system are uniform throughout space, the system itself is only the same at certain intervals in space, specifically at the lattice points, and is, therefore, less symmetrical than the empty space it occupies.

Spontaneous symmetry breaking is ubiquitous in nature, and is particularly useful for describing phase transitions in systems comprised of many particles. Much like its spatial counterpart, a time crystal was proposed as a phase of matter that spontaneously breaks *time-translation symmetry — *in effect,* *a stable equilibrium state that evolves periodically and indefinitely through time, despite the laws governing the system being invariant in time. But a time crystal that breaks continuous time-translation symmetry was proven impossible in thermal equilibrium, i.e., under true time-invariant conditions. Time, it appeared, didn’t admit the same broken symmetries as did space, in perpetual defiance of physicists’ desire for complete spacetime unity.

**Discrete Time Crystals**

Later work showed that a time crystal breaking *discrete *time*-*translation symmetry might be possible under certain conditions, however, like in driven systems. Unlike the breaking of continuous time-translation symmetry, which veers into the realm of source-less perpetual motion, the spontaneous breaking of discrete time-translation symmetry entails a system that stably oscillates in time, specifically with a different periodicity than the laws/conditions governing the system’s evolution. The breakthrough relied on engineering a carefully balanced dance between two phenomena: the many-body localization (MBL) of a one-dimensional chain of quantum spins, and the simultaneous application of a periodic driving pulse characteristic of so-called Floquet systems. I define both below.

## Many-Body Localization (MBL)

Interacting spins in a lattice will usually try to align themselves with the nearest neighboring spins, yielding the lowest possible energy configuration. At the same time, random fluctuations due to the ambient temperature tend to prevent long-range order from forming, except at extremely low temperatures. When there is sufficient disorder affecting an interacting spin lattice, however, the spins counterintuitively resist the random process of thermalization responsible for erasing quantum correlations. Instead, long-lasting stability in the spin chain occurs, an effective localization in the state-space available to the system. In principle, thermal equilibrium is never reached.

Importantly, it was discovered that the eigenstates of an MBL spin chain are “ordered” in pairs: if a given spin chain configuration is many-body localized, then so is the configuration with all the spins flipped. While such eigenstate order often arises in equilibrium quantum systems at low temperatures, it is not typical of out-of-equilibrium quantum systems that change over time, such as a periodically driven Floquet system.

## Floquet MBL-DTC

In general, Floquet systems are systems that are subject to a periodic drive, akin to a child being pushed on a swing. In the context of creating discrete time crystals (DTCs), the driven system is typically a 1D spin chain, while the driving system is a laser pulse capable of interacting with the spins.

If a many-body localized quantum spin chain is driven periodically by a pulse that flips all spins in the chain, the spins will oscillate between stable MBL states that, as previously noted, come in pairs. These oscillations exhibit considerable resistance to small changes in the driving frequency, yielding an out-of-equilibrium phase capable of persisting indefinitely and stably under all small perturbations. Even more strangely, though such oscillations occur in sub-harmonic resonance with the driving frequency (i.e., with a period that is an integer multiple of the driving period), they absorb no net energy from the driving source — otherwise, the system would be seen to gradually thermalize into a featureless, highly-disordered state. As these stable qubit oscillations occur with a period that is twice the driving period, the system spontaneously breaks the discrete time-translation symmetry of the driving pulse, thus satisfying the definition of a discrete time crystal.

## Qiskit Implementation

Now we will dive into the simulation of the MBL-DTC phase on an IBM Quantum cloud simulator. We will follow Google’s recently-published paper to create a Floquet quantum circuit that applies a Floquet pulse, simulates nearest-neighbor Ising interactions, and incorporates random single-qubit disorder, as depicted below. Another team of researchers at the University of Melbourne in Australia also created a time crystal on an IBM device back in May; you can read their paper here and see how their implementation differs from the one detailed in this blog.

First, we apply a Floquet driving pulse to each qubit that almost exactly flips all the qubits. Next, we simulate nearest-neighbor Ising interactions with random coupling values chosen from [-1.5π, -0.5π], followed by random longitudinal fields implemented using Z-rotations by a random amount in range [-π, π]. This creates the disordered interactions required for many-body localization. We implement the Floquet driving pulse as single-qubit rotations about the X axis by an amount *g*π, where *g* is a free parameter that we can vary between 0.5 and 1. If we take *g*=1, all qubits will be flipped exactly, returning to their initial state after two pulses. For *g *much smaller than 1, the qubit system occupies the thermal phase, with qubit Z-polarizations rapidly randomizing after only a few applications of the Floquet circuit (i.e., Floquet cycles).

To achieve the MBL-DTC phase, we let g~1 and randomly choose both the magnitude of coupling in the Ising interactions, which we realize as two-qubit rotation gates about ZZ, as well as the strength of the single-qubit longitudinal fields, realized as single-qubit rotations about the Z axis. The disorder in interaction and longitudinal field strengths gives rise to MBL, inducing stability in the qubit oscillations to perturbations in *g* away from 1.

The top plots show the Z-polarization of all qubits over 50 Floquet cycles, while the bottom plots show the Z-polarization of qubit 10 for the same Floquet circuit instance. When g is far from 1, the Z-polarization quickly randomizes. However, for g close to 1, the Z-polarizations of all qubits exhibit the stable and persistent oscillations characteristic of the MBL-DTC phase. Lastly, we determined the evolution of the mean Z-polarization of each qubit by applying *t* Floquet cycles followed by the projective measurement of all qubits, with *t* being varied to determine the time-dependence.

*Learn more about using IBM Quantum devices to do condensed matter physics research **here*

## References & Further Reading

- Google Quantum AI and collaborators, “Observation of Time-Crystalline Eigenstate Order on a Quantum Processor”, arXiv:2107.13571v2 [quant-ph] https://arxiv.org/pdf/2107.13571.pdf
- Randall et al., “Observation of a many-body-localized discrete time crystal with a programmable spin-based quantum simulator”, arXiv:2107.00736v1 [quant-ph], https://arxiv.org/pdf/2107.00736.pdf
- Frey and Rachel, “Realization of a discrete time crystal on 57 qubits of a quantum computer”, arXiv:2105.06632 [quant-ph] https://arxiv.org/pdf/2105.06632.pdf
- Wolchover, N. (2021, July 30). Eternal Change for No Energy: A Time Crystal Finally Made Real.
*Quanta Magazine*. https://www.quantamagazine.org/first-time-crystal-built-using-googles-quantum-computer-20210730/.