In Part II we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent.The lower (existence) bound for stabilizer codes is proved by a counting argument for classical self-orthogonal quaternary codes. Upper bounds for any quantum codes are proved by linear programming. We present two general solutions of the LP problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1.
We study the performance of quantum error correction codes (QECCs) under the detection-induced coherent error due to the imperfectness of practical implementations of stabilizer measurements, after running a quantum circuit. Considering the most promising surface code, we find that the detection-induced coherent error will result in undetected error terms, which will accumulate and evolve into logical errors. However, we show that such errors will be alleviated by increasing the code size, akin to eliminating other types of errors discussed previously. We also find that with detection-induced …
Quantum error detection has always been a fundamental challenge in a fault-tolerant quantum computer. Hence, it is of immense importance to detect and deal with arbitrary errors to efficiently perform quantum computation. Several error detection codes have been proposed and realized for lower number of qubit systems. Here we present an error detection code for a (2n+1)-qubit entangled state using two syndrome qubits and simulate it on IBM's 16-qubit quantum computer for a 13-qubit entangled system. The code is able to detect an arbitrary quantum error in any one of the first 2n qubits of the (…
A critical component of any quantum error-correcting scheme is detection of errors by using an ancilla system. However, errors occurring in the ancilla can propagate onto the logical qubit, irreversibly corrupting the encoded information. We demonstrate a fault-tolerant error-detection scheme that suppresses spreading of ancilla errors by a factor of 5, while maintaining the assignment fidelity. The same method is used to prevent propagation of ancilla excitations, increasing the logical qubit dephasing time by an order of magnitude. Our approach is hardware-efficient, as it uses a single mult…
We propose a post-selection technique, based on quantum error detection, for quantum key distribution (QKD) systems that run over quantum repeaters with encoding. In such repeaters, quantum error correction techniques are used for entanglement distillation. By developing an analytical approach to study such quantum repeaters, we show that, in the context of QKD, it is often more efficient to use the error detection, rather than the error correction, capability of the underlying code to sift out cases where an error has been detected. We implement our technique for three-qubit repetition codes …
Efficient error-mitigation techniques demanding minimal resources is key to quantum information processing. We propose a generic protocol to mitigate quantum errors using detection-based quantum autoencoders. In our protocol, the quantum data are compressed into a latent subspace while leaving errors outside, the latter of which is then removed by a measurement and post-selection. Compared to previously developed methods, our protocol on the one hand requires no extra qubits, and on the other hand has a near-optimal denoising power, in which under reasonable requirements all errors detected ou…
This paper reports on experiments realized on several IBM 5Q chips which show evidence for the advantage of using error detection and fault-tolerant design of quantum circuits. We show an average improvement of the task of sampling from states that can be fault-tolerantly prepared in the $[[4,2,2]]$ code, when using a fault-tolerant technique well suited to the layout of the chip. By showing that fault-tolerant quantum computation is already within our reach, the author hopes to encourage this approach.
In this paper, we describe and experimentally demonstrate an error detection scheme that does not employ ancilla qubits or mid-circuit measurements. This is achieved by expanding the Hilbert space where a single logical qubit is encoded using several physical qubits. For example, one possible two qubit encoding identifies $|0\rangle_L=|01\rangle$ and $|1\rangle_L=|10\rangle$. If during the final measurement a $|11\rangle$ or $|00\rangle$ is observed an error is declared and the run is not included in subsequent analysis. We provide codewords for a simple bit-flip encoding, a way to encode the …
The realization of quantum error correction is an essential ingredient for reaching the full potential of fault-tolerant universal quantum computation. Using a range of different schemes, logical qubits can be redundantly encoded in a set of physical qubits. One such scalable approach is based on the surface code. Here we experimentally implement its smallest viable instance, capable of repeatedly detecting any single error using seven superconducting qubits, four data qubits and three ancilla qubits. Using high-fidelity ancilla-based stabilizer measurements we initialize the cardinal states o…
A major milestone of quantum error correction is to achieve the fault-tolerance threshold beyond which quantum computers can be made arbitrarily accurate. This requires extraordinary resources and engineering efforts. We show that even without achieving full fault tolerance, quantum error detection is already useful on the current generation of quantum hardware. We demonstrate this experimentally by executing an end-to-end chemical calculation for the hydrogen molecule encoded in the [[4, 2, 2]] quantum error-detecting code. The encoded calculation with logical qubits significantly improves th…
We present a binary classifier based on neural networks to detect gapped quantum phases. By considering the errors on top of a suitable reference state describing the gapped phase, we show that a neural network trained on the errors can capture the correlation between the errors and can be used to detect the phase boundaries of the gapped quantum phase. We demonstrate the application of the method for matrix product state calculations for different quantum phases exhibiting local symmetry-breaking order, symmetry-protected topological order, and intrinsic topological order.
High quality, fully-programmable quantum processors are available with small numbers (<1000) of qubits, and the scientific potential of these near term machines is not well understood. If the small number of physical qubits precludes practical quantum error correction, how can these error-susceptible processors be used to perform useful tasks? We present a strategy for developing quantum error detection for certain gate imperfections that utilizes additional internal states and does not require additional physical qubits. Examples for adding error detection are provided for a universal gate se…
