Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state [1]. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state [1]. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state~\cite{Bravyi}. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Quantum error correction and fault-tolerance are at the heart of any scalable quantum computation architecture. Developing a set of tools that satisfy the requirements of fault- tolerant schemes is thus of prime importance for future quantum information processing implementations. The Clifford gate set has the desired fault-tolerant properties, preventing bad propagation of errors within encoded qubits, for many quantum error correcting codes, yet does not provide full universal quantum computation. Preparation of magic states can enable universal quantum computation in conjunction with Clifford operations, however preparing magic states experimentally will be imperfect due to implementation errors. Thankfully, there exists a scheme to distill pure magic states from prepared noisy magic states using only operations from the Clifford group and measurement in the Z-basis, such a scheme is called magic state distillation [1]. This work investigates the robustness of magic state distillation to faults in state preparation and the application of the Clifford gates in the protocol. We establish that the distillation scheme is robust to perturbations in the initial state preparation and characterize the set of states in the Bloch sphere that converge to the T-type magic state in different fidelity regimes. Additionally, we show that magic state distillation is robust to low levels of gate noise and that performing the distillation scheme using noisy Clifford gates is a more efficient than using encoded fault-tolerant gates due to the large overhead in fault-tolerant quantum computing architectures.
Abstract Magic states have been widely studied in recent years as resource states that help quantum computers achieve fault-tolerant universal quantum computing. The fault-tolerant quantum computing requires fault-tolerant implementation of a set of universal logical gates. Stabilizer code, as a commonly used error correcting code with good properties, can apply the Clifford gates transversally which is fault tolerant. But only Clifford gates cannot realize universal computation. Magic states are introduced to construct non-Clifford gates that combine with Clifford operations to achieve universal quantum computing. Since the preparation of quantum states is inevitably accompanied by noise, preparing the magic state with high fidelity and low overhead is the crucial problem to realizing universal quantum computation. In this paper, we survey the related literature in the past 20 years and introduce the common types of magic states, the protocols to obtain high-fidelity magic states, and overhead analysis for these protocols. Finally, we discuss the future directions of this field.
Magic state distillation is one of the leading candidates for implementing universal fault-tolerant logical gates. However, the distillation circuits themselves are not fault-tolerant, so there is additional cost to first implement encoded Clifford gates with negligible error. In this paper we present a scheme to fault-tolerantly and directly prepare magic states using flag qubits. One of these schemes requires only three ancilla qubits, even with noisy Clifford gates. We compare the physical qubit and gate cost of our scheme to the magic state distillation protocol of Meier, Eastin, and Knill (MEK), which is efficient and uses a small stabilizer circuit. For low enough noise rates, we show that in some regimes the overhead can be improved by several orders of magnitude compared to the MEK scheme which uses Clifford operations encoded in the codes considered in this work.
Magic states, by allowing non-Clifford gates through gate teleportation, are important building blocks of fault-tolerant quantum computation. Magic state distillation protocols aim to create clean copies of magic states from many noisier copies. However, the prevailing protocols require substantial qubit overhead. We present a distillation protocol based on permutation-invariant gnu codes, as small as two qubits. The two-qubit protocol achieves a 0.5 error threshold and 1/2 distillation rate, surpassing prior schemes for comparable codes. Our protocol furthermore distils magic states with arbitrary magic by varying the position of the ideal input states on the Bloch sphere. We achieve this by departing from the usual magic state distillation formalism, allowing the use of non-Clifford gates in the distillation protocol, and allowing the form of the output state to differ from the input state. Our protocol is compatible for use in tandem with existing magic state distillation protocols to enhance their performance.
Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely , using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill -type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately 100 × p 2 in terms of the physical error rate p . For example, with a physical error rate of p = 10 − 4 ( 10 − 3 ), the logical error rate is reduced to p L = 10 − 6 ( 10 − 4 ), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.