Craig Gidney

Improved Techniques for Preparing Eigenstates of Fermionic Hamiltonians

Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure.Quantum simulation: exponential improvement for simulating fermionic systemsImproved quantum algorithm can now simulate the electronic structure of materials and molecules much faster than what was possible before. A team involving researchers from Macquarie University, Microsoft Research, and Google has developed theoretical tools to efficiently prepare the initial “guess” for the electronic state one wants to simulate, in a way that it correctly includes the property of fermionic systems called “anti-symmetrization”, which means that the total wavefunction changes sign if two particles exchange their positions. The team also managed to reduce the number of physical operations that the quantum computer should perform to evolve the initial guess up to the final solution. Future large-scale quantum simulations should be able to model electronic structure much more efficiently than standard computers, with significant impact in chemistry, biochemistry and material science.

Halving the cost of quantum addition

We improve the number of T gates needed to perform an n-bit adder from 8n + O(1) to 4n + O(1). We do so via a temporary logical-AND construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shors algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of 8n + O(1), a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.