Quantum simulation

Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems. However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, i.e., quantum simulation. Quantum simulation promises to have applications in the study of many problems in, e.g., condensed-matter physics, high-energy physics, atomic physics, quantum chemistry, and cosmology. Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct. A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins, and photons have been proposed as quantum simulators. This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field.

Figure 1

Schematic representation of a quantum system and a corresponding quantum simulator. The quantum state $|\phi (0)⟩$ evolves to $|\phi (t)⟩$ via the unitary transformation $U=\exp \{-i\hslash H_{\text{sys}}t\}$. The quantum simulator evolves from the state $|\psi (0)⟩$ to $|\psi (t)⟩$ via $U^{\prime }=\exp \{-i\hslash H_{\text{sim}}t\}$. The simulator is designed such that there is a mapping between the simulator and the simulated system, in particular, the mappings $|\phi (0)⟩\leftrightarrow |\psi (0)⟩$, $|\phi (t)⟩\leftrightarrow |\psi (t)⟩$, and $U\leftrightarrow U^{\prime }$. While the simulated system may not be controllable (or not experimentally accessible in some cases), the quantum simulator is. Namely, the initial state $|\psi (0)⟩$ can be prepared, the unitary evolution $U^{\prime }$ can be engineered, and the final state $|\psi (t)⟩$ can be measured. The result of this measurement provides information about the simulated system. The colored arrows denote the controllable operations. The solid black arrows describe the time evolution of the system and the simulator. The dashed arrows indicate the correspondence between the quantum states of the simulator and the simulated system.

Figure 2

Initial-state preparation. Quantum circuit for the recursive procedure used to find an efficient gate sequence for preparing a given target state $|{\psi }_T(n,1)⟩$ of one electron occupying $n$ possible orbitals. The procedure uses reverse engineering, where one considers the problem of transforming the target state to the initial state ${|0⟩}^{\otimes n}$. This reverse problem allows an intuitive, systematic solution. Once the solution of this inverse problem is found, it can be inverted in order to prepare the target state $|{\psi }_T(n,1)⟩$ from the initial state ${|0⟩}^{\otimes n}$. The unitary operations $\tilde{H}$ and ${\tilde{H}}^{\prime }$ can be calculated easily from the given target state: each one of them transforms the known state of the corresponding qubit to $|0⟩$. The unitary operation $Q(n-2,1)$ transforms $|{\psi }_T(n-2,1)⟩$ into ${|0⟩}^{\otimes (n-2)}$. Adapted from [325].

Figure 3

Quantum circuit for simulating the three-body Hamiltonian $H={\sigma }_1^z\otimes {\sigma }_2^z\otimes {\sigma }_3^z$. The circuit contains six cnot gates and utilizes a fourth, ancilla qubit (bottom line) in order to achieve the desired effective Hamiltonian. From [224].

Figure 4

Quantum circuit for the calculation of molecular energies in [17]. The circuit implements the recursive phase estimation algorithm. The first iteration gives the phase $\phi$ (which represents the molecular energy) to four bits of accuracy. Each subsequent iteration incorporates the previous estimate and increases the accuracy by one bit, i.e. reduces the uncertainty by a factor of 2. Here $H$ denotes the Hadamard gate, $QF{T}^{+}$ is the inverse quantum Fourier transform, and $V_k=[\exp (-i2\pi {\phi }_{k-1}){\widehat{V}}_{k-1}{]}^2$. Adapted from [17].

Figure 5

Quantum circuits (a) for the measurement of the quantity $⟨U^{†}V⟩$ for two unitary operators $U$ and $V$, and (b) for the measurement of the spectrum of a Hermitian operator $\widehat{Q}$. Both algorithms use one ancilla qubit, which is initially prepared in the state $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$. The black dot represents a $|1⟩$-controlled gate and the white dot a $|0⟩$-controlled gate. Adapted from [286].

Figure 6

Resource requirements for the quantum simulation of the dynamics of $N$ particles interacting through a pairwise potential, maintaining a relatively small error level. The chemical symbols are a guide to show what type of problem can be simulated with a given computation size. The vertical dashed lines represent the current limit of numericallyexact quantum simulations on classical computers on a grid. From [159].

Figure 7

Trotter error analysis and gate count for a simulation of the hydrogen molecule using a DQS algorithm. (a) The calculated ground state energy of the hydrogen molecule as a function of the time step duration $\mathrm{\Delta }t$. The horizontal lines indicate the bounds for $\pm 10^{-4}{E}_h$ precision. (b) Total number of gates for a single construction of the approximate unitary as a function of $\mathrm{\Delta }t$. From [176].

Figure 8

Different systems that could implement a specialized quantum simulator for the study of problems in condensed-matter physics. Examples of such analog quantum simulators include atoms, ions, photons, nuclear and electronic spins, as well as superconducting circuits. These systems could be designed such that they form one- or two-dimensional arrays of qubits that can be manipulated in different manners. They can be thought of as toy models of the magnified lattice structure of a “solid,” with a magnification factor of a few orders of magnitude.

Figure 9

Manipulation of atoms and ions as proposed by [153]. (a) Neutral atoms in a double optical potential: the interaction between neighboring atoms is realized by displacing one of the lattices with respect to the other. Shaded circles denote the state $|1⟩$ and white ones denote the state $|0⟩$. (b) Ions in independent trapping potentials: the interaction between two neighboring ions is achieved by conditionally displacing the corresponding ions with a state-dependent force. Shaded circles denote the state $|1⟩$ and white ones denote the state $|0⟩$. Adapted from [153].

Figure 10

Polar molecules trapped in a plane by an optical potential created by two counterpropagating laser beams with wave vectors indicated by the middle arrows. The dipoles are aligned perpendicular to the plane by a dc electric field, indicated by the left arrow. The right arrow represents the ac microwave field. From [250].

Figure 11

Arrangement of electrodes for the realization of a bilayer honeycomb lattice. The microtrapping regions are shown as ellipsoids, while the locations of unwanted spurious microtraps are indicated with small spheres. From [270].

Figure 12

A one-dimensional array of cavities with atoms trapped in the cavities. Photons can hop between the different cavities because of the overlap between the light modes in adjacent cavities. The atoms can be driven using externally applied lasers. The atoms and cavities form hybrid excitations (polaritons) that can hop between the different cavities and that effectively interact with each other. The polariton-polariton interaction strength can, for example, be tuned via the atom-cavity detuning. From [134].

Figure 13

An array of quantum dots realized by a metallic gate with an array of two different size holes placed on top of an ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}/\mathrm{GaAs}$ heterostructure. A negative gate voltage is applied to this gate in order to create a potential for the two-dimensional electron gas that is otherwise free to move at the ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}/\mathrm{GaAs}$ interface. This was one of the early theoretical proposals for AQS. From [201].

Figure 14

Schematic diagram of an array of superconducting flux qubits ([16]). The T-shaped circuits are used to couple the flux qubits, while the $\phi$-shaped circuits are used to read out the states of the qubits. The gaps represent tunnel junctions.

Figure 15

A possible design for implementing the Jaynes-Cummings lattice using superconducting qubits. The colored strips are superconducting resonators and the dots are superconducting qubits. An effective particle-particle interaction is created by the qubit-cavity interactions. From [169].

Figure 16

One- or two-dimensional arrays of qubits and controls can be used to simulate various models in condensed-matter physics. Examples of such analog quantum simulators include (a) atoms in optical lattices ([152]), (b) one-dimensional or (c) two-dimensional arrays of cavities ([122]; [134]); (d) ions in linear chains, (e) two-dimensional arrays of planar traps ([71]), or (f) two-dimensional Coulomb crystals ([247]); (g) electrons in quantum dot arrays created by a mesh gate ([201]; [60]), (h) in arrays of superconducting circuits, or (i) trapped on the surface of liquid helium ([217]). The average distance between the atoms in optical lattices is less than $1\mu \mathrm{m}$. Cavity arrays based on photonic band gap cavities would also have submicron intersite separations. As for the inter-ion distances in ion trap arrays, they are about $10–15\mu \mathrm{m}$ and about the same for two-dimensional Coulomb crystals. In arrays of quantum dots, the spacing between dots is about $0.1\mu \mathrm{m}$. In superconducting circuits, the distance between qubits is typically a few microns. In the case of electrons on helium the distance between neighboring sites would be about 1 $\mu \mathrm{m}$. These interqubit distances (from 0.1 to $10\mu \mathrm{m}$) are to be compared with the far smaller average interatomic distances in solids, which are $\le 1\mathrm{nm}$. The systems shown above realize a one- or two-dimensional array of qubits which can be manipulated in different manners. The larger distances between qubits make quantum simulators more controllable and easier to measure. Therefore, they can be thought of as models of the magnified lattice structure of a solid, with a magnification factor of 2 to 4 orders of magnitude. From [59].

Figure 17

Schematic diagram of a proposal for the simulation of anisotropic spin models using polar molecules trapped in an optical lattice: a two-dimensional square lattice with nearest-neighbor orientation-dependent Ising interactions. The effective interactions between the spins $S_1$ and $S_2$ of the molecules in their rovibrational ground states are generated with a microwave field $E(t)$ inducing dipole-dipole interactions between the molecules with dipole moments $D_1$ and $D_2$, respectively. From [214].

Figure 18

Illustration of the experiment of [220], in which the states of a superconducting circuit were used to emulate a particle with spin $s$, up to $s=3/2$. For simulating a spin $s$, $2s+1$ quantum states of the circuit are used. For example, in (d) four of the circuit’s quantum states are mapped onto the four quantum states of a spin-$3/2$ particle, while a fifth state was used in the experiment as a reference state for phase measurement. By applying carefully tuned microwave-frequency drive signals, the circuit can simulate the interaction between the spin and a magnetic field with arbitrary strength and direction. An analogy with a classical system (a soccer game), shown on the left side, can also be used to give a simple explanation of the redistribution of the population among the different energy levels in terms of players passing a ball among each other, with the goalkeeper serving as the anchor. From [227].

Figure 19

Examples of analog quantum simulators of quantum phase transitions using (a) ultracold neutral atoms and (b) trapped ions. (a) The quantum phase transition from a superfluid to a Mott insulator phase realized by [123] using rubidium atoms trapped in an optical lattice. The ratio between the tunneling energy and the on-site interaction energy was controlled in [123] by adjusting the lattice potential depth, such that the quantum phase transition was observed. There are other alternative ways of simulating this quantum phase transition with arrays of cavities ([134]), or arrays of Josephson junctions ([314]). (b) Magnetic quantum phase transition simulated by [103] using trapped calcium ions. The interactions between individual spins and an external magnetic field were simulated by coupling the internal levels (representing the spin-$1/2$ states) with a resonant rf field, while the spin-spin interactions were simulated using a state-dependent optical dipole force implemented by a walking wave. From [59].

Figure 20

Spin configurations in a triangular lattice and their experimental signatures, simulated using the motional degrees of freedom of atoms trapped in an optical lattice, averaged over several experimental runs. The coupling parameters can be tuned to ferromagnetic (solid lines) or antiferromagnetic values (dashed lines), and these determine the resulting spin configuration. From [294].

Figure 21

An example of the simulation of the simplest case of spin frustration: (a) three antiferromagnetically coupled spins. This simulation was realized with (b) three ytterbium ions in a one-dimensional trap. $J_1$ is the nearest-neighbor coupling and $J_2$ is the next-nearest-neighbor coupling. (c) The expected form of the Ising interactions $J_1$ and $J_2$, controlled through the laser detuning $\tilde{\mu }$, used to generate an optical spin-dependent force, is depicted. The three oscillation modes shown at the top of (c) are used as markers, as the parameter $\tilde{\mu }$ is defined relative to their frequencies. From [165].

Figure 22

Schematic diagram of the basic building block of a Kitaev lattice, consisting of four superconducting charge qubits (labeled 1 to 4): (i) Qubits 1 and 2 are inductively coupled via a mutual inductance $M$; (ii) qubits 1 and 3 are coupled via an $LC$ oscillator; and (iii) qubits 1 and 4 are capacitively coupled via a mutual capacitance $C_m$. Inset: These three types of interqubit couplings are denoted as $X$, $Y$, and $Z$ bonds. From [360].

Figure 23

(a) Proposed setup for the simulation of ring-exchange models . The bosons (black dots) reside on a square lattice with the molecules (gray dots) at the centers of the plaquettes. Adapted from [56]. (b) Simulation of compact U(1) lattice gauge theories. Dipolar bosons reside on the sites of the kagome lattice (black dots); the hexagonal dual lattice is the lattice formed by the centers of the corner-sharing triangles (gray dots). Adapted from [304].

Figure 24

Sisyphus cooling and amplification in a superconducting circuit ([121]; [226]). The two energy levels of a superconducting qubit are shown as a function of the current applied to the circuit. The current in a nearby superconducting resonator drives the qubit, and a microwave drive signal is applied to the qubit. The cycle $a\to b\to c\to d$ on the right side mimics Sisyphus cooling in atomic physics: the energy of emitted photons is higher than the energy of photons absorbed from the drive signal, causing a constant removal of energy from the resonator. In the analogy with Sisyphus in Greek mythology the current in the resonator plays the role of Sisyphus and the qubit plays the role of the rock. Sisyphus is constantly pushing the rock uphill (as can be seen in steps a and c), and the rock keeps going back to the bottom of the hill. The cycle $e\to f\to g\to h$ on the left side represents the opposite scenario, Sisyphus amplification, where energy is constantly added to the resonator. Adapted from [226].