Expanded version of quant-ph/9906114 for Proceedings of special session at Jan., 2000 AMS meeting. Includes more background on spatial wave functions, and a proof (by H. Pollatsek) that the permutationally invariant 9-bit code is non-additive
In many physically realistic models of quantum computation, Pauli exchange interactions cause a special type of two-qubit errors, called exchange errors, to occur as a first order effect of couplings within the computer. We discuss the physical mechanisms behind exchange errors and codes designed to explicitly deal with them.
The performance of quantum error correction can be significantly improved if detailed information about the noise is available, allowing to optimize both codes and decoders. It has been proposed to estimate error rates from the syndrome measurements done anyway during quantum error correction. While these measurements preserve the encoded quantum state, it is currently not clear how much information about the noise can be extracted in this way. So far, apart from the limit of vanishing error rates, rigorous results have only been established for some specific codes. In this work, we rigorously…
Spatiotemporal Pauli processes: Quantum combs for modeling correlated noise in quantum error correction
Monash University（莫纳什大学）；Agency for Science, Technology and Research (A*STAR)（科学技术研究局（A*STAR））；CSIRO（澳大利亚联邦科学与工业研究组织）；The University of Melbourne（墨尔本大学）；Singapore University of Technology and Design（新加坡科技设计大学）；National University of Singapore（新加坡国立大学）
In quantum error correction, the description of noise channel cannot be completely accurate, and fluctuation always appears in noise channel. It is found that when fluctuation of physical noise channel is considered, the average effective channel is dependent only on the average of physical noise channel, and the average of physical noise channel here plays the role of the independent error model in the previous works. Now, one may conclude that in the independent error model, the results in previous works are also valid for average channel where fluctuation exists. In some typical cases, our …
The intrinsic probabilistic nature of quantum systems makes error correction or mitigation indispensable for quantum computation. While current error-correcting strategies focus on correcting errors in quantum states or quantum gates, these fine-grained error-correction methods can incur significant overhead for quantum algorithms of increasing complexity. We present a first step in achieving error correction at the level of quantum algorithms by combining a unified perspective on modern quantum algorithms via quantum signal processing (QSP). An error model of under- or over-rotation of the si…
Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here we prove several results on efficiently learning Pauli channels, and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on $n$ qubits with high probability to a relative precision $ε$ using $O\bigl(ε^{-2} n 2^n\bigr)$ measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measu…
It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group coho…
One of the main problems in quantum information systems is the presence of errors due to noise, and for this reason quantum error-correcting codes (QECCs) play a key role. While most of the known codes are designed for correcting generic errors, i.e., errors represented by arbitrary combinations of Pauli X , Y and Z operators, in this paper we investigate the design of stabilizer QECC able to correct a given number eg of generic Pauli errors, plus eZ Pauli errors of a specified type, e.g., Z errors. These codes can be of interest when the quantum channel is asymmetric in that some types of err…
Semiclassical trajectory-coherent approximation in quantum mechanics: II. High order corrections to the Dirac operators in external electromagnetic field
Color code thresholds under circuit-level noise beyond the Pauli framework
INFN, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；Institute for Complex Quantum Systems, Ulm University（乌尔姆大学复杂量子系统研究所）；Padua Quantum Technologies Research Center, Università degli Studi di Padova（帕多瓦大学帕多瓦量子技术中心）
The error threshold of a one-parameter family of quantum channels is defined as the largest noise level such that the quantum capacity of the channel remains positive. This in turn guarantees the existence of a quantum error correction code for noise modeled by that channel. Discretizing the single-qubit errors leads to the important family of Pauli quantum channels; curiously, multipartite entangled states can increase the threshold of these channels beyond the so-called hashing bound, an effect termed superadditivity of coherent information. In this work, we divide the simplex of Pauli chann…
Protecting Quantum Information Encoded in Decoherence Free States Against Exchange Errors
在量子信息处理器中，相同量子比特之间的交换相互作用会产生酉双量子比特错误。本文证明了集体消相干的消相干自由子空间（DFS）在交换条件下会发生泡利误差，但不会使消相干自由态脱离DFS。为了保护DFS免受这些错误的影响，采用最近提出的级联DFS-量子纠错码方案就足够了[D. A. Lidar，D.培根和K.B. Whaley，Phys. Rev. Lett. { \bf 82}，4556（1999）]。
The Clifford group is a finite subgroup of the unitary group generated by the Hadamard, the CNOT, and the Phase gates. This group plays a prominent role in quantum error correction, randomized benchmarking protocols, and the study of entanglement. Here we consider the problem of finding a short quantum circuit implementing a given Clifford group element. Our methods aim to minimize the entangling gate count assuming all-to-all qubit connectivity. First, we consider circuit optimization based on template matching and design Clifford-specific templates that leverage the ability to factor out Pau…
Measurement-based quantum computing (MBQC) promises natural compatibility with quantum error correcting codes at the cost of a polynomial increase in physical qubits. MBQC proposals have largely focused on photonic systems, where 2-qubit gates are difficult. Semiconductor spin qubits in quantum dots, on the other hand, offer fast 2-qubit gates via the exchange interaction. In exchange-based quantum computing, as with other solid-state qubits, leakage to higher states is a serious problem that must be mitigated. Here, two hybrid measurement-exchange schemes are proposed which quantify the benef…
The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being …
