Quantum Computing with Arrays of Atoms
From Materials Simulation to Quantum Machine Learning
Unlike classical information carried by digital bits 0 or 1, quantum information is commonly encoded onto a collection of two-level quantum systems referred to as qubits. A register containing n qubits is described by a large complex vector of dimension 2ⁿ. By taking advantage of the exponential amount of information available in quantum registers, several quantum algorithms have been proposed that could outperform state-of-the-art classical algorithms for specific computing tasks. In order to build a viable quantum computing device, a broad variety of physical platforms are currently being investigated [1]. Among them, arrays of single neutral atoms appear as a very powerful and scalable technology to manipulate quantum registers with up to a few hundreds of qubits [2].
In such quantum processors, each qubit is encoded into two electronic states of an atom. Nature provides that all the qubits are strictly identical when taken independently, a remarkable advantage to achieve low error rates during the computation. A typical cycle consists in four main phases: register preparation, sub-register ordering, quantum processing and register readout. All these tasks can be performed with very high fidelities by coupling the atoms to laser light fields.
Operation an atomic qubit register
The first technological challenge is to assemble and readout a qubit register made of hundreds of individual atoms. A promising solution based on arrays of optical tweezers [3] uses the hardware components shown in Figure 1a.
A. Register loading
As a starting point, a dilute atomic vapor is formed inside an ultra-high vacuum system operated at room temperature. With a first laser system (not shown), a cold ensemble of about 10⁶ atoms and 1mm³ volume is prepared inside a magneto-optical trap, leveraging numerous laser cooling and trapping techniques [4]. Then, a second trapping laser system isolates individual atoms within this ensemble. Using a high numerical aperture lens, the trapping beam gets strongly focused down to multiple spots of about 1µm diameter: the so-called optical tweezers [5]. With a trapping volume of a few µm³, each tweezer contains at most one single atom at a time.
The number of tweezers and their arrangement in 1D, 2D or even 3D is fully tailored by holographic methods [6]. Before passing through the lens, the trapping beam is reflected onto a spatial light modulator (SLM) that imprints an adjustable phase pattern on the light. In the focal plane of the lens, the phase modulation gets converted into a desired intensity pattern, as illustrated in Fig. 1b. For this reason, the neutral atom platform for quantum computing has a unique potential for scalability: the size of the quantum register is only limited by the amount of trapping laser power and by the performance of the optical system generating the optical tweezers.
B. Realizing a defect-free sub-register
Although each tweezer can host at most one atom, it happens only in about 50% of the cases, the tweezer being otherwise empty. To detect which of the tweezers are filled, the atoms are imaged by collecting their fluorescence onto a sensitive camera (see Fig. 1a). The atoms are then moved from site to site in order to generate a pre-defined sub-register with unit filling. This operation is done using fast and programmable moving optical tweezers. Elementary rearrangement steps are described in Fig. 2a. From the analysis of the initial image, an algorithm computes on the fly a set of moves to rearrange the initial configuration into the desired fully assembled sub-register. After rearrangement is completed, an assembled image is acquired to confirm the new positions of the atoms in the sub-register.
C. Quantum processing and register readout
Once the register is fully assembled, the quantum computation task can start. In the circuit-model approach of quantum computing, information is processed by applying a sequence of quantum logic gates to the register. The next section gives more details on the realization of those quantum logic gates.
After the last gate, the atomic register is readout by taking a final fluorescence image. It is performed such that each atom in qubit state |0⟩ will appear as bright, whereas atoms in qubit state |1⟩ remain dark, as illustrated in Fig. 2b. Given the probabilistic nature of each possible outcome imposed by quantum mechanics, such computation cycles are then repeated many times in order to reconstruct the relevant statistical properties of the final quantum state produced by the algorithm.
Quantum logic gates with atomic qubits
Computation is implemented using quantum logic gates. The spacing between atoms in the register being typically several micrometers, addressability with high fidelity is possible with optical methods. Quantum logic gates are therefore realized by shining fine-tuned laser pulses onto a chosen subset of single atoms in the register. These control laser pulses drive transitions between electronic levels, which modify the state of the chosen qubits. Interestingly, one- and two-qubits gates are all that is needed to constitute a universal set allowing to run any quantum algorithm. We describe here how they are implemented in a neutral atom register.
A. One-qubit gates
One-qubit gates are specific unitary transformations described by 2-by-2 complex matrices transforming one qubit state into another. Notable examples are the NOT-gate that changes the state |0⟩ into |1⟩ and vice-versa, or the Hadamard (H) gate that generates superposition of both states starting from a pure state. In the {|0⟩,|1⟩} basis, theses gates read
Any one-qubit gate can be implemented by driving the atomic qubit transition with a control field using well-chosen duration τ, Rabi frequency Ω (the strength of the driving, controlled by the intensity of the laser pulse) and detuning δ (the difference between the qubit transition and the laser field frequencies). Regarding the previous examples, the NOT-gate is obtained by pulsing the control field at δ = 0 with a pulse area Ω τ equals to π, and the H-gate at δ = Ω with a pulse area also equals to π.
B. Two-qubit gates based on Rydberg interactions
Two-qubit gates are unitary transformations described by 4-by-4 matrices that transform one two-qubit state into another. They are the most basic but crucial resource allowing to generate entanglement in the register. Physically, their implementation requires an interaction between the qubits. However, neutral atoms in their electronic ground state can only interact significantly via contact collisions. Therefore, single atoms — typically separated by a few micrometers in the register — do not naturally feel each other.
In 2000, Jaksch et al. [7] proposed a scheme that takes advantage of the dipole-dipole interaction between atoms when they are prepared to highly excited electronic states. In these so-called Rydberg states, the atoms exhibit a huge electric dipole moment that can be three orders of magnitude bigger than in their ground state. Therefore, two Rydberg atoms — separated by a few micrometers — will experience a dipole-dipole interaction strong enough to shift significantly the energy of the doubly excited state, preventing the excitation of two atoms at the same time. This effect is called Rydberg blockade and is the basic mechanism to achieve a quantum logic: the excitation of a first atom to a Rydberg state conditions the excitation of a second one.
C. Example: Implementation of the CNOT-gate
Rydberg-mediated entanglement is particularly well suited to implement the controlled-NOT (CNOT) gate. This operation flips a “target” qubit state if and only if a “control” qubit is in the state |1⟩. The corresponding matrix in the pair state basis {|0c 0t⟩,|0c 1t⟩,|1c 0t⟩,|1c 1t⟩} of control and target qubits is
There are several ways to realize this gate using dipolar Rydberg interaction [8]. Figure 3a illustrates the key mechanism with a simple sequence of 3 pulses:
- With initial state |1c 1t⟩, all the three pulses are off-resonant, and the state remains unchanged.
- With initial state |0c 1t⟩, pulse 2 is off-resonant so the target qubit state remains unchanged. Pulses 1 and 3 drive a 2π rotation of the control qubit. The pair state picks up a phase factor exp(iπ)= -1.
- With initial state |1c 0t⟩, pulses 1 and 3 are off-resonant so the control qubit state remains unchanged. Pulse 2 drives a 2π rotation of the target qubit. The pair state picks up a phase factor exp(iπ)= -1
- With initial state |0c 0t⟩, pulse 1 excites the control qubit to state |r⟩ with a π rotation. Because of the Rydberg blockade, the target qubit state |r⟩ is shifted out of resonance during pulse 2. The target qubit remains in state |0t⟩. Finally, pulse 3 brings the control qubit back to state |0c⟩ with another π rotation. The pair state picks up a factor exp(iπ) = -1.
This pulse sequence realizes, within a global π phase, the controlled-Z (CZ) gate:
The CZ gate can then be used to generate the CNOT gate by including Hadamard gates on the target atom before and after the operation, as shown in Fig. 3b.
From universal computing to simulation and optimization
Although it is theoretically possible to implement any quantum circuit with only one- and two-qubit gates, it is not optimal when the required number of such gates becomes large and their imperfections are taken into account. Practical realizations can benefit a lot from the ability to implement quantum gates acting on more than two qubits at a time, a key resource to shorten circuit length through quantum compilation. In that respect, atomic quantum processors based on Rydberg-gates are extremely promising. Indeed, dipolar interactions naturally arise between all the excited Rydberg atoms and the blockade mechanism can be made efficient onto the whole register. It is therefore possible to implement multi-qubit gates rather simply in comparison to most other physical systems [9].
In the limit of involving the whole register, a large logic gate can also be seen as a global quantum Hamiltonian. This approach is known as quantum simulation since it allows to mimic complex systems from materials science in the lab with higher level of control. But it is not restricted to the description of quantum many-body systems. Some of those Hamiltonians can significantly help solving classical optimization problems. For instance, Ising-type Hamiltonians are suitable to implement an algorithm finding the maximum independent set in large graphs [10]. The versatility of the neutral atoms platform also opens promising opportunities to implement hybrid quantum-classical convolutional neural networks [11] that will potentially yield a quantum advantage.
The Authors
Lucas Béguin and Adrien Signoles are Quantum Engineers in charge of Hardware at Pasqal, a full stack Quantum Computing startup powered by arrays of neutral atoms.
To learn more
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