Rydberg Atoms/Quantum simulation

Simulating the time evolution of a quantum many-body system on a classical computer is believed to be a task that requires resources that grow exponentially with the size of the system. The underlying lies in the tensor product structure of the composite Hilbert space, meaning that for every $$d$$-level system added to the problem, the number of basis states in the system increases by a factor of $$d$$. While there are sophisticated concepts, e.g., based on matrix product states, these only work in specific instances and an efficient simulation of quantum many-body dynamics on classical computers is not possible.

In 1982, Richard Feynman realized that in order to overcome this exponential scaling, the simulation device itself would have to be governed by quantum mechanics, from which the idea of quantum simulation was born. Building a quantum simulator requires to have excellent control over the degrees of freedom represented by the quantum device. This has led to two different routes for designing a quantum simulation. One possibility is to build a specific device that is exactly undergoing the same Hamiltonian evolution as the simulated system. Such devices are called analog quantum simulators. For instance, a many-body system of laser driven Rydberg atoms in the limit of small blockade radius could be called a quantum simulator of the Ising model. The other possibility is to construct a device that is, in principle, capable to simulate the dynamics of any other quantum system. Similar to classical computers, this works best when the simulation dynamics becomes discrete, i.e., the simulator reproduces the state of the simulated system only at certain timesteps $$k\tau$$, while in between the state of the simulator can be completely different. Such an approach is known as digital quantum simulator. For practical purposes, we restrict the definition of such a universal quantum simulator to the simulation of Hamiltonians with short-ranged interactions, as all physical interactions reduce to purely local interaction at some point. Similar to simulations on classical computers, it is also helpful to introduce abstraction layers so that we can use suitable approximations for the inner workings of the quantum simulator. If we decide to ignore the actual physical implementation for now, the lowest level we can consider is given by quantum logic gates, which have originally been discussed in the context of quantum computing

Quantum logic gates
Quantum logic gates are represented by unitary matrices that transform the quantum state before the operation into another after the application of the quantum gate. As such, they can be seen as the time-evolution operator acting for discrete timesteps,
 * $$|\psi(\tau_{n+1})\rangle = U(\tau_n,\tau_{n+1})|\psi(\tau_n)\rangle.$$

The basis set for the quantum state $$|\psi\rangle$$ is given by a product basis of two-level systems (quantum bits or "qubits"),
 * $$|\psi\rangle = \sum_{i \in \{0,1\}, j \in \{0,1\},\ldots} c_{ij\ldots} |ij\ldots\rangle.$$

We can also think of the operation $$U$$ to be constructed out of several smaller building blocks,
 * $$U(\tau_n,\tau_{n+1}) = \prod_i^N U(\tau_{n+(i-1)/N},\tau_{n+i/N}).$$

For simplicity, we want to restrict ourselves to a universal set of quantum gates that can be used to construct any other gate from it. This can be realized by a set of three quantum gates, including the $$z$$ rotation gate,
 * $$R_z(\phi) = \exp(i\phi\sigma_z) = \left(\begin{array}{cc}e^{i\phi} & \\ & e^{-i\phi}\end{array}\right),$$

where $$\phi$$ is an arbitrary rotation angle. To construct any other single qubit quantum gate, we need a second gate that does not commute with $$R_z$$. The most convenient choice is the Hadamard gate given by
 * $$U_H = \frac{1}{\sqrt{2}}\left(\begin{array}{cc}1 & 1\\1 & -1\end{array}\right).$$

The Hadamard gate can be used to transform $$\sigma_z$$ into $$\sigma_x$$ and vice versa, i.e.,
 * $$R_x(\phi) = \exp(i\phi\sigma_x) = \left(\begin{array}{cc} \cos \phi & i\sin\phi \\ i\sin\phi & \cos\phi\end{array}\right) = U_H R_z(\phi) U_H$$
 * $$R_z(\phi) = U_H R_x(\phi) U_H.$$

Note that while the Hadamard gate is Hermitian, $$U_H^\dagger = U_H$$, most quantum gates are not. Finally, rotations about the $$y$$ axis can be constructed as
 * $$R_y(\phi) = \exp(i\phi\sigma_y) = R_z(-\pi/4)R_x(\phi)R_z(\pi/4).$$

Single qubit rotation do not allow us to generate entanglement between the qubits. Therefore, it is necessary to include a two-qubit quantum gate, which is most coveniently chosen as the controlled-not (CNOT) gate, acting on two qubits $$A$$ and $$B$$ as
 * $$U_{CNOT} = |0\rangle\langle 0|_A \otimes 1_B + |1\rangle\langle 1|_A \otimes \sigma^x_B = \left(\begin{array}{cccc}1\\ & 1\\& & & 1\\ & & 1\end{array}\right).$$

Its function can be understood as follows: If the "control" qubit $$A$$ is in the 0 state, nothing happens. However, if $$A$$ is in 1, the target qubit $$B$$ gets flipped by the $$\sigma_x$$ operation. As an example, let us study the creation of entanglement between two qubits by a gate sequence consisting of a single Hadamard gate, followed by a CNOT operation. The qubits are initialized in the product state $$|00\rangle$$. Then, we have
 * $$|\psi\rangle = U_{CNOT} U_H^A |00\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle),$$

which is a maximally entangled state as its reduced density matrix is the maximally mixed state. The set of $$R_z(\phi)$$, $$U_H$$, and $$U_{CNOT}$$ forms a universal set for all $$N$$-qubit quantum gates.

It is often convenient to use a pictorial representation for quantum gate networks, also known as quantum circuits. Each qubit is represented by a straight line, with the horizontal axis denoting time, and each single qubit gate is shown a rectangular box acting on the particular qubit. Two-qubit gates are represented in a similar way, with the control qubit being indicated by a small circle, see Fig. 1. In this notation, much more complex quantum circuits can be represented and analyzed, for instance the decomposition of the Toffoli gate,
 * $$U_T = \left(\begin{array}{cccccccc} 1 \\ & 1\\ & & 1\\ & & & 1\\ & & & & & 1\\ & & & & & & & 1\\ & & & & & & 1\end{array}\right),$$

which is a 3-qubit variant of the CNOT gate and is universal for all classical computations. It can be constructed from elementary gates, see Fig. 2, where $$T = R_z(\pi/8)$$.



Digital simulation procedure
Suppose we want to simulate a four-body spin interaction of the form
 * $$H=E_0 \sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}\sigma_x^{(4)}.$$

This four-body spin operator has two eigenvalues, $$\pm 1$$, which are eightfold degenerate. The key idea is to use an additional auxiliary particle and encode the eigenvalue into its spin state. If the auxiliary control spin is initially in $$|0\rangle$$, this can be done using the gate sequence
 * $$G = U_H^{(c)}\left(\prod_{i=1}^4 U_{CNOT}^{(c,i)}\right) U_H^{(c)}.$$

To understand this in more detail, let us look at the behavior of the gate sequence on the spin state $$|\pm 1,\lambda\rangle$$, where $$\lambda$$ labels the state within the degenerate manifold. The first Hadamard gate will yield
 * $$U_H^{(c)}|0\rangle_c |\pm 1,\lambda\rangle = \frac{1}{\sqrt{2}}(|0\rangle_c+|1\rangle_c)|\pm 1,\lambda\rangle.$$

Applying the sequence of CNOT gates will multiply the eigenvalue of the spin interaction, conditional on the control spin being in $$|1\rangle$$,
 * $$\prod_i U_{CNOT}^{c,i} \frac{1}{\sqrt{2}}(|0\rangle_c+|1\rangle_c)|\pm 1,\lambda\rangle = \frac{1}{\sqrt{2}}(|0\rangle_c\pm|1\rangle_c)|\pm 1,\lambda\rangle.$$

Finally, the second Hadamard gate will give us
 * $$U_H^{(c)}\frac{1}{\sqrt{2}}(|0\rangle_c+|1\rangle_c)|+1,\lambda\rangle = |0\rangle_c|+1,\lambda\rangle$$
 * $$U_H^{(c)}\frac{1}{\sqrt{2}}(|0\rangle_c-|1\rangle_c)|-1,\lambda\rangle = |1\rangle_c|-1,\lambda\rangle$$.

Consequently, we have mapped the eigenvalue of the four-body interaction operator onto the state of a single auxiliary spin.

The full quantum simulation of the dynamics $$U=\exp(-iHt)$$ can then be realized by applying a $$z$$ rotation to the control spin and reverse the mapping $$G$$,
 * $$U = \exp(-iE_0\sigma_x^{(1)}\sigma_x^{(2)}\sigma_x^{(3)}\sigma_x^{(4)}t) = G R_z(-\phi) G.$$

The phase of the $$z$$ rotation is related to the timescale of the simulation according to $$\phi = E_0 t$$.

For a many-body system, the full dynamics can be simulated if the gate sequences are applied in parallel (if they act on independent spins) or sequentially (if they act on the same spins). However, in case of non-commuting operators, one has to ensure that this sequential operations does not introduce errors. This is true if the timestep $$\tau$$ of the simulation procedure is sufficiently small, as can be seen from the Suzuki-Trotter expansion
 * $$\exp[-i(H_A+H_B)\tau] = \exp(-iH_A\tau)\exp(-iH_B\tau) + O(\tau^2).$$

This completes the toolbox required for the realization of a universal quantum simulator.