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Author: Norman Yao
Nearly four decades ago, Richard Feynman gave a visionary lecture, “Simulating Physics with Computers,” in which he emphasized the impossible complexity of simulating a quantum mechanical system using a classical computer (Feynman 1960, 1982). Indeed, even describing the full quantum state of ~60 electron spins is well beyond current computational capabilities. To overcome this challenge, Feynman proposed the notion of a “quantum simulator.”
The intuition is strikingly simple: Make use of fully controllable quantum building blocks to mimic the interactions that underlie a less accessible quantum system (Buluta and Nori 2009; Lloyd 1996). Experimental progress in this direction has been truly extraordinary, making it possible to isolate single microscopic particles (at the nanoscale), to manipulate and control their internal quantum states, and to detect them with almost perfect fidelity (Cirac and Zoller 2012; Georgescu et al. 2014).
This paper describes three of the major experimental platforms associated with quantum simulation—ultracold atomic systems (Bloch et al. 2012), polar molecules (Moses et al. 2017), and superconducting quantum bits (qubits) (Houck et al. 2012)—and gives examples of the phenomena they can simulate.
Many-Body Phases in Ultracold Atomic Systems
Ultracold quantum gases provide a number of unique opportunities when it comes to simulating nature. They offer a tremendous amount of control, can be imaged with single-atom resolution, and can mimic the underlying structure of solid state materials (Bloch et al. 2012; Cirac and Zoller 2012). In addition, perhaps the most crucial aspect underlying their broad scientific impact is the existence of a flexible array of cooling techniques that can quench the kinetic energy of atomic systems. Indeed, ultracold atomic systems have actually reached subnanokelvin temperatures, revealing phenomena ranging from Bose-Einstein condensation and Cooper-paired superfluidity to Mott insulators and localization.
Despite these successes, the temperature of atomic quantum simulations is still too high to simulate a number of more exotic—and delicate—quantum mechanical phases, including antiferromagnetic spin liquids, fractional Chern insulators, and high-temperature superconductivity. The figure of merit for observing such physics is not the absolute temperature but rather the dimensionless entropy density.
Reaching ultralow entropy densities remains a major challenge for many-body quantum simulations despite the multitude of kinetic cooling techniques. This challenge is particularly acute for gases in deep optical lattice potentials, for which transport, and thus evaporative cooling, is slowed. Moreover, in lattice systems representing models of quantum magnetism, the entropy resides primarily in spin, rather than motional, degrees of freedom. Expelling such entropy through evaporative cooling requires the conversion of spin excitations to kinetic excitations, a process that is typically inefficient.
Two broad approaches have been proposed to overcome this challenge. The first is adiabatic preparation: one initializes a low entropy state and changes the Hamiltonian gradually until the desired many-body state is reached. However, the final entropy density is bounded from below by the initial entropy density, and experimental constraints or phase transitions may preclude a suitable adiabat. The second approach is to “shift entropy elsewhere” (Stamper-Kurn 2009) using the system’s own degrees of freedom as a bath. This approach helps to stabilize the Mott-insulating phase of the Bose-Hubbard model, where the low-density wings of the system serve as an entropy sink, allowing for in situ evaporative cooling.
On the applications front, ultracold atomic simulations have raised the possibility of studying topological phases in out-of-equilibrium spin systems. Unlike traditional condensed matter systems, one cannot simply “cool” to a desired topological ground state by decreasing the temperature of a surrounding bath. Rather, preparation must proceed coherently. This necessitates both detailed knowledge of the phase transitions separating topological states from their short-range-entangled neighbors and understanding of the interplay of topology, lattice -symmetries, and out-of-equilibrium dynamics.
One particular context where lattice and topology meet is in the notion of fractional Chern insulators–exotic phases, which (as explained in Yao et al. 2013) “arise when strongly interacting particles inhabit a flat topological band structure. Particles injected into these exotic states of matter fractionalize into multiple independently propagating pieces, each of which carries a fraction of the original particle’s quantum numbers. While similar effects underpin the fractional quantum Hall effect observed in continuum two dimensional electron gases, fractional Chern insulators, by contrast, are lattice dominated. They have an extremely high density of correlated particles whose collective excitations can transform nontrivially under lattice symmetries.” Since the full configuration interaction state generally competes with superfluid and crystalline orders, the resulting phase diagram exhibits both conventional and topological phases.
Spin Liquids in Polar Molecules
Polar molecules trapped in optical lattices have recently emerged as a powerful new platform for quantum simulation (Moses et al. 2017). This platform exhibits many advantages, including local spatial addressing, stable long-lived spins, and intrinsic long-range dipolar interactions. Typically, the molecules are subject to a static electric field and their motion is pinned by a strong laser field (figure 1). This implies that the degree of freedom, which (often) participates in the quantum simulation, is an effective rotational excitation.
These rotational excitations can simulate a large number of interesting many-body quantum phases. In particular, by varying the DC electric field strength as well as the tilt of the electric field vector, one can sharply modify the geometry of the dipoles and introduce additional dispersion into their single-particle band structures. At the same time, increasing the electric field strength enhances the long-range interactions. These qualitative differences in the microscopics of polar molecules yield a rich phase diagram exhibiting both conventional and topological phases, including crystalline ordering, superfluids, and chiral spin liquids. The nature of these phases can be characterized using diagnostics such as the ground-state degeneracy and the real-space structure factor.
Of particular interest in the context of polar molecular simulations is the realization of quantum spin -liquids. These are characterized by entanglement over macroscopic scales and can exhibit a panoply of exotic properties, ranging from emergent gauge fields and fractionalized excitations to robust chiral edge modes.
Recent work has demonstrated that polar molecule simulations naturally realize the so-called dipolar Heisenberg antiferromagnet. Such simulation requires only a judicious choice of two molecular rotational states (to represent a pseudo-spin) and a constant electric field. The simplicity of this system stems from the use of rotational states with no angular momentum about the electric field axis and contrasts with previous works where nonzero matrix elements appear for the transverse electric dipole operator, unavoidably generating ferromagnetic spin-spin interactions. Motivated by this physical construction, large-scale numerical studies of the dipolar Heisenberg model (e.g., Yan et al. 2011) find evidence for quantum spin liquid ground states on both triangular and Kagome lattices.
Quantum Walks in Superconducting Qubits
Much like their classical stochastic counterparts, -discrete-time quantum walks have stimulated activity across a broad range of disciplines. In the context of computation, they provide exponential speedup for certain oracular problems and represent a universal platform for quantum information processing. -Quantum walks also exhibit features characteristic of diverse physical phenomena and thus are an ideal platform for quantum simulation.
It has recently been demonstrated that quantum walks can be directly realized using superconducting transmon qubits coupled to a high-quality-factor electro-magnetic cavity (Flurin et al. 2017; figure 2). The quantum walk takes place in the phase space of the cavity mode and each lattice site corresponds to a particular coherent state of the cavity, while the two logical states of the superconducting qubit form the internal spin of the walker. Coherent spin rotations can be performed using microwave driving, while spin-dependent translations arise naturally from the dispersive coupling between the qubit and the cavity.
A unique application of this particular quantum simulation platform is in the direct measurement of so-called topological invariants. In these protocols, a geometric signature of the topological invariant is imprinted as a Berry phase on the quantum state of the particle; the phase can then be extracted and dis-entangled from -other contributions via a simple interferometric protocol.
The quantum simulation community has made remarkable progress in the controlled manipulation of individual quanta (e.g., Lanyon et al. 2011; Simon et al. 2011). These advances have opened the door for the engineering of quantum many-body systems as well as the development of quantum technologies.
Looking forward, the continued dialogue between atomic, molecular, and optical physics, condensed matter physics, and quantum information science promises to be fruitful for both the fundamental and applied sciences, enabling the simulation of macroscopic quantum behavior and providing detailed microscopic intuition.
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 “Exotic” in this context refers to two types of phases of matter: (1) those that have been theoretically predicted to exist but have not yet been observed in nature and (2) those that have been observed but are not yet fully understood.