We present a single framework to describe parity codes in exclusively spatial dimensions as quantum error correction codes, as well as parity codes with a temporal dimension, as quantum circuits. This framework builds on the stabilizer formalism to introduce label tracking of parity information for illustrating and constructing quantum circuits and quantum error correction codes. Additionally, this framework proposes the necessary transformations and conditions to translate between the spatial and temporal versions of a given parity code. This technique is then applied to find the temporal variant of the spatially extended LHZ-code [1] using CNOT gates on a linear nearest-neighbor (LNN) qubit architecture. The resulting quantum circuit provides an efficient implementation of quantum algorithms on LNN architectures, that require the interactions found in all-to-all connected transverse-field Ising models. Two widely used quantum algorithms that require such interactions are the Quantum Approximate Optimization Algorithm (QAOA) [2] and the quantum Fourier transform (QFT). For the QAOA, this is the first proposal with a linear circuit depth while not requiring more than$n^{2}$CNOT gates per iteration. Our approach for QFT, requiring only LNN connectivity, surpasses previous QFT implementations even on devices with high qubit connectivity in terms of circuit depth while not increasing the gate count in leading order.
We present a single framework to describe parity codes in exclusively spatial dimensions as quantum error correction codes, as well as parity codes with a temporal dimension, as quantum circuits. This framework builds on the stabilizer formalism to introduce label tracking of parity information for illustrating and constructing quantum circuits and quantum error correction codes. Additionally, this framework proposes the necessary transformations and conditions to translate between the spatial and temporal versions of a given parity code. This technique is then applied to find the temporal variant of the spatially extended LHZ-code [1] using CNOT gates on a linear nearest-neighbor (LNN) qubit architecture. The resulting quantum circuit provides an efficient implementation of quantum algorithms on LNN architectures, that require the interactions found in all-to-all connected transverse-field Ising models. Two widely used quantum algorithms that require such interactions are the Quantum Approximate Optimization Algorithm (QAOA) [2] and the quantum Fourier transform (QFT). For the QAOA, this is the first proposal with a linear circuit depth while not requiring more thann2CNOT gates per iteration. Our approach for QFT, requiring only LNN connectivity, surpasses previous QFT implementations even on devices with high qubit connectivity in terms of circuit depth while not increasing the gate count in leading order.
One of the main challenges the implementation of quantum optimization algorithms has been facing in the last year is the connectivity of qubits on quantum chips. The parity architecture has emerged as an approach to address this issue by trading qubit connectivity for an overhead in the number of physical qubits by encoding the parity of logical qubits. This qubit overhead comes with so-called parity constraints that can be implemented geometrically locally on the quantum chip. While the parity architecture is suitable for both analog and digital quantum computing, we focus on the digital side in this thesis, and present new insights on the parity approach from two perspectives. First, we investigate its resource requirements for quantum optimization, in particular for the Quantum Approximate Optimization Algorithm (QAOA), and propose a constructive algorithm to reduce these requirements compared to previous strategies. In addition, by viewing the parity constraints as stabilizers, we introduce a strategy to conserve the parity constraints throughout the execution of the QAOA. This is in contrast to previous work on the parity architecture, where the parity constraints were typically enforced via additional energy penalties. Second, we show how the parity framework can be used for universal quantum computation, and how cornerstone algorithms like the quantum Fourier transform benefit from this new approach. Our method essentially allows to trade a high circuit-depth for a qubit overhead, while at the same time offering more flexibility in circuit design by enabling on-the-fly recoding of physical qubits. Furthermore, we present a universal quantum computing approach in the parity architecture based on quantum measurements, which allows for additional reductions in circuit depth. Finally, we outline how the parity architecture can be exploited for quantum error correction. With these findings, we aim to contribute to building the bridge between the current era of non-error-corrected hardware and fault-tolerant quantum computing.
We present a formalism based on tracking the flow of parity quantum information to implement algorithms on devices with limited connectivity without qubit overhead, SWAP operations, or shuttling. Instead, we leverage the fact that entangling gates not only manipulate quantum states but can also be exploited to transport quantum information. We demonstrate the effectiveness of this method by applying it to the quantum Fourier transform (QFT) and the quantum approximate optimization algorithm (QAOA) with n qubits. This improves upon all state-of-the-art implementations of the QFT on a linear nearest-neighbor architecture, resulting in a total circuit depth of 5 n − 3 and requiring n 2 − 1 gates. For the QAOA, our method outperforms SWAP networks, which are currently the most efficient implementation of the QAOA on a linear architecture. We further demonstrate the potential to balance qubit count against circuit depth by implementing the QAOA on twice the number of qubits using bilinear connectivity, which approximately halves the circuit depth.
We present a protocol to encode and decode arbitrary quantum states in the parity architecture with constant circuit depth using measurements, local nearest-neighbor and single-qubit operations only. While this procedure typically requires a quadratic overhead of simultaneous qubit measurements, it allows for a simple and low-depth implementation of logical multi-qubit gates in the parity encoding via code deformation. We discuss how such encoding and decoding schemes can be used to flexibly change the size and shape of the underlying code to enable a more efficient implementation of quantum gates or algorithms. We apply the new findings to the QAOA which leads to a constant depth implementation using local gates at the same optimization performance as the standard, potentially non-local, QAOA approach without the parity encoding. Furthermore, we show that our method can reduce the depth of implementing the quantum Fourier transform by a factor of two when allowing measurements.
Understanding the dynamics of interacting quantum systems is a central goal of quantum simulation. For geometrically local Hamiltonians, locality suggests that digital simulation should retain much of the parallelism of the underlying dynamics. For a target accuracy ε, we construct a nearest-neighbor quantum circuit that simulates the dynamics of an n-qubit, time-independent, finite-range Hamiltonian with normalized local interaction strength on a fixed-dimensional lattice for time T≥1, with circuit depth O(Tlog(nT/ε)) and gate count O(nTlog(nT/ε)). This reduces the polylogarithmic overhead of Haah, Hastings, Kothari, and Low to a single logarithm. On logarithmic-scale blocks from the Lieb--Robinson decomposition, a shallow but highly accurate implementation is essential for reducing the circuit depth. Our key observation is that spatial localization also controls the cost of high precision. In each block, the error of a constant-order product formula is compensated for by a correction, yielding exponential accuracy. The correction can be implemented with depth proportional to the block diameter, introducing no additional asymptotic depth scale beyond that required by spatial localization.
Furthermore, we prove an unconditional circuit-depth lower bound for uniform XXZ Hamiltonians: at fixed evolution time T in the stated regime and sufficiently small fixed error ε, any nearest-neighbor quantum circuit simulating these dynamics requires depth Ω(logn/loglogn). This matches our O(logn) upper bound up to a doubly logarithmic factor, establishing near-optimal circuit depth for lattice Hamiltonian simulation in this regime. Together, these results clarify the role of locality and precision in determining the circuit complexity of quantum dynamics and suggest broader principles for spatially constrained quantum computation.
Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis∣0⟩and∣1⟩states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis∣0⟩and∣1⟩states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)[[9,5,3]]3,[[5,1,3]]3, and[[7,1,3]]3codes, and a 9–21% reduction for the ququint (d = 5)[[10,6,3]]5code when compared to prior work. We also achieved circuit depth reductions up to 42%.
We propose a versatile method for implementing quantum algorithms that simultaneously minimizes gate count and circuit depth. Central to our approach are Parity Twine chains-CNOT-based building blocks tailored to hardware connectivity. This technique surpasses existing state-of-the-art methods in executing key quantum algorithms, including the Quantum Fourier Transform and the Quantum Approximate Optimization Algorithm, across various quantum hardware architectures such as linear chains, square grids, hexagonal lattices, ladders, and fully connected systems.