Towards quantum chemistry on a quantum computer

UQ (University of Queensland, Brisbane Australia) physicists and Harvard chemists have teamed up to build a quantum computer that could have profound implications for wider science.

Professor Andrew White and colleagues from UQ’s School of Mathematics and Physics, teamed up with researchers from Harvard University, led by Professor Alán Aspuru-Guzik, to tackle the problem of applying quantum mechanics to fields such as chemistry and biology.

“Physicists have a problem,” Professor White said.

“They have an outstandingly successful theory of nature at the small scale – quantum mechanics – but have been unable to apply it exactly to situations more complicated than, say, four or five atoms.

“But now we have done exactly that by building a small quantum computer and used it to calculate the precise energy of molecular hydrogen.”

This groundbreaking approach to molecular simulations could have profound implications not just for chemistry, but also for a range of fields from cryptography to materials science.

The work, published this week in Nature Chemistry, saw Professor White’s team assemble the physical computer and run the experiments, while Professor Aspuru-Guzik’s team coordinated experimental design and performed key calculations.

“We were the software guys and they were the hardware guys,” Professor Aspuru-Guzik said.

While modern supercomputers can perform approximate simulations, increasing the complexity of these systems results in an exponential increase in computational time.

“Quantum computers promise highly precise calculations while using a fraction the resources of conventional computing,” he said.

“This computational power derives from the way quantum computers manipulate information. In classical computers, information is encoded in bits, that have only two values: zero and one. Quantum computers use quantum bits – qubits – that can have an infinite different number of values such as zero, or one, or zero plus one, and so on.

“Quantum computers also exploit the strange phenomena of entanglement, powerful correlations between qubits that Einstein once described as ‘spooky action at a distance’.”

Professor White said it would be a while before quantum computers would leave the lab and appear on desktops.

“It’s very early days for quantum technology,” he said.

“Most quantum computer demonstrations have been limited to a handful of qubits. A colleague of mine in Canada says that any demonstration with less than ten qubits is cute but useless, which makes me think of a baby with an abacus.

“However, Alán and his team at Harvard have shown that when we can build circuits of just a few hundred qubits, this will surpass the combined computing power of all the traditional computers in the world, each of which uses many billions of bits.”

“It took standard computing 50 years to get to this point, I’m sure we can do it in much less time than that.”

The quantum circuits corresponding to evolution of the listed Hermitian
second-quantized operators. Here p, q, r, and s are orbital indices corresponding to qubits such that the population of |1 determines the occupancy of the orbitals. It is assumed that the orbital indices satisfy p > q > r > s. These circuits were found by performing the Jordan-Wigner transformation given in (S2b) and (S2a) and then propagating the obtained Pauli spin variables. In each circuit, θ = θ(h) where h is the integral preceding the operator. Gate Tˆ(θ) is defined by ˆ T |0 = |0 and ˆ T |1 = exp(−iθ)|1, ˆG is the global phase gate given by exp(−iφ)ˆ1, and the change-of-basis gate ˆ Y is defined as ˆRx(−π/2). Gate ˆH refers to the Hadamard gate. For the number-excitation operator, both M = ˆ Y and M = ˆH must be implemented in succession. Similarly, for the double excitation operator each of the 8 quadruplets must be implemented in succession. The global phase gate must be included due to the phase-estimation procedure. Phase estimation requires controlled versions of these operators which can be accomplished by changing all gates with θ-dependence into controlled gates.

Nature Chemistry – Towards quantum chemistry on a quantum computer

Exact first-principles calculations of molecular properties are currently intractable because their computational cost grows exponentially with both the number of atoms and basis set size. A solution is to move to a radically different model of computing by building a quantum computer, which is a device that uses quantum systems themselves to store and process data. Here we report the application of the latest photonic quantum computer technology to calculate properties of the smallest molecular system: the hydrogen molecule in a minimal basis. We calculate the complete energy spectrum to 20 bits of precision and discuss how the technique can be expanded to solve large-scale chemical problems that lie beyond the reach of modern supercomputers. These results represent an early practical step toward a powerful tool with a broad range of quantum-chemical applications.

12 page pdf of supplemental information

A fundamental challenge for the quantum simulation of large molecules is the accurate decomposition of the system’s time evolution operator, ˆU . In our experimental demonstration, we exploit the small size and inherent symmetries of the hydrogen molecule Hamiltonian to implement ˆU exactly, using only a small number of gates. As the system size grows such a direct decomposition will no longer be practical. However, an efficient first-principles simulation of the propagator is possible for larger chemical systems.

The key steps of an efficient approach are: (1) expressing the chemical Hamiltonian in second quantized form, (2) expressing each term in the Hamiltonian in a spin 1/2 representation via the Jordan-Wigner transformation, (3) decomposing the overall unitary propagator, via a Trotter-Suzuki expansion, into a product of the evolution operators for non-commuting Hamiltonian terms, and (4) efficiently simulating the evolution of each term by designing and implementing the corresponding quantum circuit. We note that the first two steps generate a Hamiltonian that can be easily mapped to the state space of qubits. The last steps are part of the quantum algorithm for simulating the time-evolution operator, ˆU , generated by this Hamiltonian. Details of each step are provided.

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Their “iterative phase estimation algorithm” is a variation on existing quantum algorithms such as Shor’s algorithm, which has been successfully used to crack encryption schemes. It is run several times in succession, with the output from each run forming the input to the next.

“You send two things into the algorithm: a single control qubit and a register of qubits pre-encoded with some digital information related to the chemical system you’re looking at,” says White.

“The control qubit entangles all the qubits in the register so that the output value – a 0 or 1 – gives you information about the energy of the chemical system.” Each further run through the algorithm adds an extra digit.

The data passes through the algorithm 20 times to give a very precise energy value. “It’s like going to the 20th decimal place,” White says. Errors in the system can mean that occasionally a 0 will be confused with a 1, so to check the result the 20-step process is repeated 30 times.

The team used this process to calculate the energy of a hydrogen molecule as a function of its distance from adjacent molecules. The results were astounding, says White. The energy levels they computed agreed so precisely with model predictions – to within 6 parts in a million – that when White first saw the results he thought he was looking at theoretical calculations. “They just looked so good.”

IPEA success probability measured over a range of parameters. Probabilities
for obtaining the ground state energy, at the equilibrium bond length 1.3886 a0, as a function of: (a) the number of times each bit is sampled (n); (b) the number of extracted bits (m); (c) the fidelity between the encoded register state and the ground state (F ). The standard fidelity between a measured mixed ρ and ideal pure |Ψ> state is F =. (a) & (b) employ a ground state fidelity of F ≈ 1. (a) & (c) employ a 20-bit IPEA. All lines are calculated using a model that allows for experimental imperfections. This model, as well as the technique used to calculate success probabilities and error bars, are detailed in the SOM (section B).

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