Flexible layout of surface code computations using AutoCCZ states

TL;DR

This work introduces the AutoCCZ state, a self-correcting CCZ resource with embedded delayed-choice CZs, enabling more flexible spacetime layouts for surface-code computations. By combining a compact delayed-choice CZ, improved CCZ distillation, and routing strategies, the authors demonstrate reaction-limited ripple-carry adder and Clifford-limited QROM reads that can operate efficiently on large-scale superconducting qubit platforms. Under plausible hardware assumptions, they find that circuit-level Toffoli parallelism provides no speedup for problems smaller than about five million physical qubits, and they achieve a fourfold reduction in delayed-choice CZ spacetime volume relative to prior approaches. The results advance practical layout design for large-scale quantum arithmetic and lookups, with explicit factory counts and footprint estimates illustrating scalability considerations.

Abstract

We construct a self-correcting CCZ state (the "AutoCCZ") with embedded delayed choice CZs for completing gate teleportations. Using the AutoCCZ state we create efficient surface code spacetime layouts for both a depth-limited circuit (a ripply-carry addition) and a Clifford-limited circuit (a QROM read). Our layouts account for distillation and routing, are based on plausible physical assumptions for a large-scale superconducting qubit platform, and suggest that circuit-level Toffoli parallelism (e.g. using a carry-lookahead adder instead of a ripple-carry adder) will not reduce the execution time of computations involving fewer than five million physical qubits. We reduce the spacetime volume of delayed choice CZs by a factor of 4 compared to techniques from previous work (Fowler 2012), and make several improvements to the CCZ magic state factory from (Gidney 2019).

Figures (18)

• Figure 1: Equivalent concepts expressed in quantum circuit diagrams, ZX calculus graphs, and 3d topological diagrams. In circuit diagrams, time goes from left to right. In ZX calculus graphs, there is no preferred time direction. In 3d topological diagrams, times goes from bottom to top. We never show Pauli operations in lattice surgery diagrams or in ZX calculus graphs, because they are performed by the classical control system instead of by operating on qubits. Our usage of the ZX calculus is somewhat non-standard in that we consider ZX graphs to be equivalent if they are equal modulo Pauli operations, we use a non-standard node coloring that matches the coloring of our topological diagrams, and we introduce a "delayed choice node" to represent adaptive effects coming from the classical control system. We exaggerate the spacing of our 3d topological diagrams, as in gidney2018magic, so that it is possible to see how the components are interconnected.
• Figure 2: Delayed choice multiplex/demultiplex construction from fowler2012time. Top left is a circuit diagram directly from fowler2012time. The bottom left shows the process as a ZX calculus graph which, unlike the circuit diagram, is identical for multiplexing and demultiplexing. Known rewrite rules are used to show equivalence with the claimed "choose which route is connected" functionality. On the right side is a 3d topological diagram of a lattice surgery implementation of the construction. The vertical poles coming out of the branches of the fork are the routing qubits used to control which of the two branches connects to the rear trunk. They can be extended arbitrarily. The red squares atop the routing qubit columns are placeholders for an eventual X or Z basis measurement, to be determined by classical control software.
• Figure 3: Our optimized delayed choice CZ as a circuit and as a lattice surgery construction. The two forms are shown to be equivalent via ZX graph identities. During execution, the choice of whether or not to apply the CZ is delayed by extending the red-topped columns (the "routing qubits") in the 3d topological diagram. Once the choice is known, the columns are terminated with the red square replaced either by a white square (activates the CZ) or a black square (skips the CZ). The circuit can be opened in the online simulator Quirk by https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B1%2C1%2C%22%E2%80%A2%22%2C%22Z%22%5D%2C%5B%22Amps2%22%5D%2C%5B%5D%2C%5B%22%E2%80%A2%22%2C1%2C%22X%22%5D%2C%5B1%2C%22%E2%80%A2%22%2C1%2C%22X%22%5D%2C%5B1%2C1%2C1%2C1%2C%22Measure%22%5D%2C%5B1%2C1%2C%22H%22%2C%22H%22%2C%22%E2%97%A6%22%5D%2C%5B1%2C1%2C%22Measure%22%2C%22Measure%22%5D%2C%5B1%2C1%2C%22%3C%3C2%22%2C1%2C%22%E2%80%A2%22%5D%2C%5B1%2C%22Z%22%2C1%2C%22%E2%80%A2%22%5D%2C%5B%22Z%22%2C1%2C%22%E2%80%A2%22%5D%2C%5B%22Amps2%22%2C1%2C1%2C1%2C%22%E2%97%A6%22%5D%2C%5B%5D%2C%5B%22Amps2%22%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%5D%2C%22init%22%3A%5B%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%5D%7D.
• Figure 4: A circuit diagram and ZX calculus graph simplification showing how to create and consume an AutoCCZ magic state, powered by delayed choice CZs, to perform a CCZ gate. The bottom left part of the circuit is producing the AutoCCZ magic state, then the middle left does parity measurements vs the target qubits, then the middle right uses those measurements to determine the basis of measurements that determine whether fixup operations occur, then finally on the right side all the measurement results are used to update the Pauli frame tracked in the control software. The circuit can be opened in the online simulator Quirk by https://algassert.com/quirk#circuit=%7B%22cols%22%3A%5B%5B1%2C1%2C1%2C%22%E2%80%A2%22%2C1%2C1%2C%22%E2%80%A2%22%2C1%2C1%2C%22Z%22%5D%2C%5B1%2C1%2C1%2C1%2C%22 cz%22%2C1%2C%22 cz%22%2C1%2C%22 cz%22%2C1%2C%22 cz%22%5D%2C%5B1%2C1%2C1%2C%22 cz%22%2C1%2C%22 cz%22%2C1%2C%22 cz%22%2C1%2C%22 cz%22%5D%2C%5B1%2C1%2C1%2C%22%E2%80%A2%22%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C%22Z%22%5D%2C%5B%22Amps3%22%5D%2C%5B%5D%2C%5B%22%E2%80%A2%22%2C1%2C1%2C%22X%22%5D%2C%5B1%2C%22%E2%80%A2%22%2C1%2C1%2C1%2C1%2C%22X%22%5D%2C%5B1%2C1%2C%22%E2%80%A2%22%2C1%2C1%2C1%2C1%2C1%2C1%2C%22X%22%5D%2C%5B1%2C1%2C1%2C%22Measure%22%2C1%2C1%2C%22Measure%22%2C1%2C1%2C%22Measure%22%5D%2C%5B1%2C1%2C1%2C1%2C1%2C1%2C%22Swap%22%2C1%2C1%2C%22Swap%22%5D%2C%5B1%2C1%2C1%2C%22Swap%22%2C1%2C1%2C%22Swap%22%5D%2C%5B1%2C1%2C1%2C%22%E2%80%A2%22%2C%22H%22%2C%22H%22%5D%2C%5B1%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%2C%22H%22%2C%22H%22%5D%2C%5B1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%2C%22H%22%2C%22H%22%5D%2C%5B1%2C1%2C1%2C1%2C%22Measure%22%2C%22Measure%22%2C1%2C%22Measure%22%2C%22Measure%22%2C1%2C%22Measure%22%2C%22Measure%22%5D%2C%5B1%2C1%2C1%2C%22%E2%80%A2%22%2C%22%3C%3C2%22%5D%2C%5B1%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%2C%22%3C%3C2%22%5D%2C%5B1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%2C%22%3C%3C2%22%5D%2C%5B%22Z%22%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B1%2C%22Z%22%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B1%2C%22Z%22%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B1%2C1%2C%22Z%22%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B1%2C1%2C%22Z%22%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B%22Z%22%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B1%2C1%2C%22Z%22%2C1%2C1%2C1%2C%22%E2%80%A2%22%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B%22Z%22%2C1%2C1%2C%22%E2%80%A2%22%2C1%2C1%2C1%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B1%2C%22Z%22%2C1%2C%22%E2%80%A2%22%2C1%2C1%2C%22%E2%80%A2%22%5D%2C%5B%22Amps3%22%5D%5D%2C%22gates%22%3A%5B%7B%22id%22%3A%22 cz%22%2C%22name%22%3A%22CZ%22%2C%22circuit%22%3A%7B%22cols%22%3A%5B%5B%22%E2%80%A2%22%2C%22Z%22%5D%5D%7D%7D%5D%2C%22init%22%3A%5B%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%2C%22%2B%22%5D%7D.
• Figure 5: Lattice surgery implementation of the CZ fixups bubble from \ref{['fig:auto-ccz-circuit']}. The 3d model is stored in ancillary file "ccz-fixup.skp"; it can be opened online using Sketchup. The tee-junctions in the 3d diagram have been numbered, so that it is easier to see how they correspond to the cycle in the ZX calculus graph. The pink box on the right, with routing qubit "chimneys", is the simplified abstract representation that we will use in larger diagrams.