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Paper 1
Flagging the Clifford hierarchy:~Fault-tolerant logical $\fracπ{2^l}$ rotations via measuring circuit gauge operators of non-Cliffords
Shival Dasu, Ben Criger
- Year
- 2026
- Journal
- arXiv preprint
- DOI
- arXiv:2603.24573
- arXiv
- 2603.24573
We provide a recursively defined sequence of flag circuits which will detect logical errors induced by non-fault-tolerant $R_{\overline{Z}}(\fracπ{2^l})$ gates on CSS codes with a fault distance of two. As applications, we give a family of circuits with $O(l)$ gates and ancillae which implement fault-tolerant logical $R_{Z}(\fracπ{2^l})$ or $R_{ZZ}(\fracπ{2^l})$ gates on any $[[k + 2, k, 2]]$ iceberg code and fault-tolerant circuits of size $O(l)$ for preparing $|\fracπ{2^l}\rangle$ resource states in the $[[7,1,3]]$ code, which can be used to perform fault-tolerant $R_{\overline{Z}}(\fracπ{2^l})$ rotations via gate teleportation, allowing for implementations of these gates that bypass the high overheads of gate synthesis when $l$ is small relative to the precision required. We show how the circuits above can be generalized to $π( x_0.x_{1}x_{2}\ldots x_{l}) = \sum_{j}^{l} π\frac{x_j}{2^j}$ rotations with identical overheads in $l$, which could be useful in quantum simulations where time is digitized in binary. Finally, we illustrate two approaches to increase the fault-distance of our construction. We show how to increase the fault distance of a Cliffordized version of the T gate circuit to $3$ in the Steane code and how to increase the fault-distance of the $\fracπ{2}$ iceberg circuit to $4$ through concatenation in two-level iceberg codes. This yields a targeted logical $R_{\overline{Z}}(\fracπ{2})$ gate with fault distance $4$ on any row of logical qubits in an $[[(k_2+2)(k_1+2), k_1k_2, 4]]$ code.
Open paperPaper 2
Quantum Simulation of Nonlinear Dynamical Systems Using Repeated Measurement
Joseph Andress, Alexander Engel, Yuan Shi, Scott Parker
- Year
- 2024
- Journal
- arXiv preprint
- DOI
- arXiv:2410.03838
- arXiv
- 2410.03838
We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations (ODEs), which may be generated from partial differential equations in plasma physics. We map a dynamical system to a Hamiltonian form, where the Hamiltonian matrix is a function of dynamical variables. To advance in time, we measure expectation values from the previous time step, and evaluate the Hamiltonian function classically, which introduces stochasticity into the dynamics. We then perform standard quantum Hamiltonian simulation over a short time, using the evaluated constant Hamiltonian matrix. This approach requires evolving an ensemble of quantum states, which are consumed each step to measure required observables. We apply this approach to the classic logistic and Lorenz systems, in both integrable and chaotic regimes. Out analysis shows that solutions' accuracy is influenced by both the stochastic sampling rate and the nature of the dynamical system.
Open paper