Quantum error correction is vital for implementing universal quantum computing. A key component is the encoding circuit that maps a product state of physical qubits into the encoded multipartite entangled logical state. Known methods are typically not ``optimal'' either in terms of the circuit depth (and therefore the error burden) or the specifics of the target platform, i.e., the native gates and topology of a system. This work introduces a variational compiler for efficiently finding the encoding circuit of general quantum error-correcting codes with given quantum hardware. Focusing on the noisy intermediate-scale quantum regime, we show how to systematically compile the circuit following an optimizing process seeking to minimize the number of noisy operations that are allowed by the noisy quantum hardware or to obtain the highest fidelity of the encoded state with noisy gates. We demonstrate our method by deriving efficient encoders for logic states of the five-qubit code and the seven-qubit Steane code. We describe ways to augment the discovered circuits with error detection. Our method is applicable quite generally for compiling the encoding circuits of quantum error-correcting codes.
Quantum error correction is vital for implementing universal quantum computing. A key component is the encoding circuit that maps a product state of physical qubits into the encoded multipartite entangled logical state. Known methods are typically not ``optimal'' either in terms of the circuit depth (and therefore the error burden) or the specifics of the target platform, i.e., the native gates and topology of a system. This work introduces a variational compiler for efficiently finding the encoding circuit of general quantum error-correcting codes with given quantum hardware. Focusing on the noisy intermediate-scale quantum regime, we show how to systematically compile the circuit following an optimizing process seeking to minimize the number of noisy operations that are allowed by the noisy quantum hardware or to obtain the highest fidelity of the encoded state with noisy gates. We demonstrate our method by deriving efficient encoders for logic states of the five-qubit code and the seven-qubit Steane code. We describe ways to augment the discovered circuits with error detection. Our method is applicable quite generally for compiling the encoding circuits of quantum error-correcting codes.
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%.
To implement quantum algorithms on a quantum computer, we must overcome the twin problems of fault-tolerance -- how can we realize a relatively noiseless computation by cleverly combining noisy components? -- and compilation -- how can we realize an arbitrary quantum algorithm given the basic operations available on the quantum device at hand? We show how treating the former problem via error-correcting codes enables greater flexibility in resolving the latter. Specifically, we explicitly leverage the fact that error-correcting codes introduce redundancy which renders physically distinct operators logically indistinguishable. In terms of computation, it suffices to implement any operator logically equivalent to some target, yet from a compilation perspective, certain choices may be preferable to others. Our novel contribution is making this intuition precise in the general setting of the special unitary group. In particular, we describe how to reduce the problem of making a compilation-ideal choice to a least squares problem and provide a closed form solution thereof. Using our framework, it is possible to circumvent inserting costly swaps to adhere to hardware connectivity; instead, we could realize the logical target through a distinct physical Hamiltonian that is natively accessible. We elucidate our approach using the [[4,2,2]] code. We discuss connections to compressed sensing that may pave the way to efficient compilation leveraging physical degrees of freedom.
The development of quantum computers requires not only experimental advances, but also theoretical efforts to tackle the bottlenecks and identify potential applications. This thesis addresses some essential elements in each stage of the development of quantum computers, from the currently available hardware to the long-term devices. The critical factor that hinders the quantum hardware from performing perfect operations is the presence of noise, however, it can be overcome by error correction codes. We first consider a large-scale quantum processor with asymmetrical noise and preferably a modular structure. By concatenating the surface code with an error-detecting code, and utilising the information from detection with a customised decoder, we find very high thresholds for the designed devices. The second topic is small error correction codes which are more realistic to be implemented in the near future. A measure `integrity' is introduced to benchmark the performance of quantum memories to demonstrate beneficial error correction, associated with four milestones. We show integrity is closely related with other commonly-used measures and can be easily assessed by experimentalists. This framework is then studied in detail in the context of an ion-trap quantum device implemented with the seven qubit code. We present a protocol based on detailed sequences of crystal-reconfiguration operations and stabiliser mappings to perform a full error correction cycle. Realistic error models and parameters from reported works are also given. While experimental efforts continue to battle against the noise, some hybrid quantum-classical algorithms have been developed to be implemented on the near-term noisy devices. The variational algorithm is one of the hybrid algorithms that have been widely applied to simulations of condensed matter physics and quantum chemistry. In the next chapter, we introduce a variational quantum compiler to automatically search for the encoding circuits of small error correction codes. By tailoring the ansatz to optimise the circuit for particular hardware systems, we show some novel circuits discovered with the compiler. The final chapter applies the variational algorithm to solve linear algebra problems including linear systems of equations and matrix-vector multiplications. We show that the linear algebra problems can be solved with near-term devices, with circuit depth linear to the condition number and super-linear to the number of qubits.
Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correction code (QECC), with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this notoriously difficult challenge, we push the limits of the depth of superconducting quantum circuits and experimentally realise the universal five-qubit QECC, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the five-qubit QECC for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of 57.1(3)% while with a high fidelity of 98.6(1)% in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of 97.2(2)% within the code space. Finally, we realise the decoding circuit and recover the input state with a process fidelity of 57.4(7)%. After decoding, by identifying errors via the measurement of the four ancillae and then mathematically recovering the qubit state, we verify the power of error correction of this code. Thus, by demonstrating each key aspect of error correction with the five-qubit code, our work establishes the viability of experimental quantum error correction with superconducting qubits and paves the route to fault-tolerant quantum computing.
Fault-tolerant quantum computers rely on Quantum Error-Correcting Codes (QECCs) to protect information from noise. However, no single error-correcting code supports a fully transversal and therefore fault-tolerant implementation of all gates required for universal quantum computation. Code switching addresses this limitation by moving quantum information between different codes that, together, support a universal gate set. Unfortunately, each switch is costly-adding time and space overhead and increasing the logical error rate. Minimizing the number of switching operations is, therefore, essential for quantum computations using code switching. In this work, we study the problem of minimizing the number of code switches required to run a given quantum circuit. We show that this problem can be solved efficiently in polynomial time by reducing it to a minimum-cut instance on a graph derived from the circuit. Our formulation is flexible and can incorporate additional considerations, such as reducing depth overhead by preferring switches during idle periods or biasing the compilation to favor one code over another. To the best of our knowledge, this is the first automated approach for compiling and optimizing code-switching-based quantum computations at the logical level.
Abstract As quantum processors grow in scale and reliability, the need for efficient quantum gate decomposition of circuits to a set of specific available gates, becomes ever more critical. The decomposition of a particular algorithm into a sequence of these available gates is not unique. Thus, the fidelity of an algorithm’s implementation can be increased by choosing an optimized decomposition. This is true both for noisy intermediate-scale quantum platforms as well as for implementation of quantum error correction schemes. Here we present a compilation scheme which implements a general-circuit decomposition to a sequence of Ising-type, long-range, multi-qubit (MQ) entangling gates, that are separated by layers of single qubit rotations. We use trapped ions as an example in which MQ gates naturally arise, yet any system that has connectivity beyond nearest-neighbors may gain from our approach. We evaluate our methods using the quantum volume (QV) test over N qubits. In this context, our method replaces 3 N 2 / 2 two-qubit gates with 2 N + 1 MQ gates. Furthermore, our method minimizes the magnitude of the entanglement phases, which typically enables an improved implementation fidelity, by using weaker driving fields or faster realizations. We numerically test our compilation and show that, compared to conventional realizations with sequential two-qubit gates, our compilations improves the logarithm of QV by 20% to 25%.