Toward all-to-all connectivity with expressive gate sets and compilation
Novel implementation of XY(θ) unlocks a new family of entangling gates
By Deanna Abrams
Near term quantum computers offer the possibility of solving useful real-world problems on qubits whose gate fidelities are still below error-correcting thresholds. The algorithms being developed to solve these real-world problems are often limited by circuit depth, which refers to the number of gates you can perform on your qubits before the outcome is dominated by noise. This problem can be tackled from two directions; either by decreasing the error of each gate operation or by decreasing the number of gates needed to enact any given algorithm. Today we’re introducing a new family of gates on our Quantum Cloud Services platform that enable dramatic decreases in gate count for near-term hybrid algorithms.
A set of qubit operations typically must be built using gates that are calibrated for a particular quantum system. The more types of calibrated gates available to use as building blocks, the fewer total gates are necessary for a given operation. And since algorithms with fewer gates means fewer errors, these programs tend to achieve better results.
Let’s consider building arbitrary operations out of the specific two-qubit gates: controlled-Z (CZ), which adds a conditional phase to a two-qubit state, and iSWAP, which exchanges quantum states between qubits, effectively shuttling quantum information across the chip. (We also need single-qubit gates, but given their much lower error rates we’ll treat them as perfect operations). Any two-qubit gate can be expressed with at most 3 CZs or 3 iSWAPs. This means that for general gates, iSWAP doesn’t offer any compilation advantages over CZ. However, it can still be useful to have both, because iSWAPs can be compiled using a single iSWAP instead of more than one CZ, and vice versa. You also get a small boost for the generic SWAP gate, which is important when mapping problems with highly connected topologies onto physical systems with limited connectivity.
Even more powerful advantages are possible if, in addition to CZ, we introduce a whole new family of two-qubit gates, parameterized by the angle θ, which we’ll write as XY(θ) (where XY(𝜋) = iSWAP). Using CZ and XY(θ), an average gate depth reduction of 32% for a random circuit is possible, counting only two-qubit entangling gates. For particular algorithms, these additional gates are more explicitly useful because they preserve qubit excitations.
Superpositions of qubit states can be thought of as combinations of 0’s and 1’s with some relative phase between them. Preserving those relative phases during computation is key to the power of quantum algorithms. CZ gates add a conditional phase to certain states, but they don’t change the phase of the states relative to each other, making it trivial to keep track of all the phases. XY gates, on the other hand, partially or fully swap qubit states and phases, which means exact bookkeeping is needed to make sure the relative phases of different qubits are all properly maintained throughout several channels of our control system. We architected a custom control system to manage the timing of pulses (and thus the phases of the pulses) we use to enact entangling gates, which made it much easier to do this bookkeeping and leverage the efficiencies of the XY gate family.
In our system, the XY interaction is a parametric gate, where the relevant parameter is the entangling angle of the gate. That is, when we tune up the XY interaction, we actually make a family of entangling gates available, which we write down as XY(θ). By allowing the user to vary θ smoothly between 0 and 2𝜋, we can control the amount of swapping among qubits. When θ = 0 or 2𝜋, the qubit states are not swapped at all. When θ = 𝜋, the qubit states are swapped exactly, and we recover the iSWAP gate. And when θ is some value between 0 and 𝜋, the qubit states have been partially swapped. Even adding just a single gate from the XY family of gates would allow users to perform algorithms using fewer gates. Having access to the full family of XY gates provides even more building blocks for creating complex circuits, leading to larger possible reductions in gate depth for many algorithms.
To unlock the whole gate family, we implemented our XY(θ) gates as two separate pulses that each enact half of the gate. By changing the phase of the pulse used to enact the second half of the gate, we can change which XY gate we implement without having to recalibrate any other parameters used to control the gate. After calibrating an XY gate between two qubits on the QCS Aspen-4 chip, we measured the fidelity of that gate for 100 randomly chosen angles, and observed a range of 95.67 ± 0.60% to 99.01 ± 0.15% fidelity, with a median fidelity of 97.35 ± 0.17%.
The utility of XY gates can be illustrated with the MaxCut QAOA problem, which seeks to partition a graph of vertices and edges into two sets such that the number of edges in each set is maximized. A common issue with MaxCut is that the graph that you want to use to solve the problem may not match the topology of the device. This often requires many SWAP gates to get distant qubits to interact. Using the XY(𝜋) gate instead allows the quantum computer to behave as if it is fully connected — that is, with a smart compiler, like quilc, a given MaxCut problem on an all-to-all connected graph will use the same number of two qubit gates no matter the actual connectivity of the device. These potentially large savings in gate count can boost the power of MaxCut-type algorithms on quantum computers with restricted qubit connectivity.
Deanna M. Abrams, Nicolas Didier, Blake R. Johnson, Marcus P. da Silva, Colm A. Ryan. Implementation of the XY interaction family with calibration of a single pulse. https://arxiv.org/abs/1912.04424v1
