We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver's half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender's side. We discuss the advantages and disadvantages for each of the two codes.
Several emerging quantum technologies, including quantum networks, modular and fusion-based quantum computing, rely crucially on the ability to perform photonic Bell state measurements. Therefore, photon losses and the 50\% success probablity upper bound of Bell state measurements pose a critical limitation to photonic quantum technologies. Here, we develop protocols that overcome these two key challenges through logical encoding of photonic qubits. Our approach uses a tree graph state logical encoding, which can be produced deterministically with a few quantum emitters, and achieves near-dete…
Liquid phase NMR is a general purpose test-bed for developing methods of coherent control relevant to quantum information processing. Here we extend these studies to the coherent control of logical qubits and in particular to the unitary gates necessary to create entanglement between logical qubits. We report an experimental implementation of a conditional logical gate between two logical qubits that are each in decoherence free subspaces that protect the quantum information from fully correlated dephasing.
We demonstrate coherent control of two logical qubits encoded in a decoherence free subspace (DFS) of four dipolar-coupled protons in an NMR quantum information processor. A pseudo-pure fiducial state is created in the DFS, and a unitary logical qubit entangling operator evolves the system to a logical Bell state. The four-spin molecule is partially aligned by a liquid crystal solvent, which introduces strong dipolar couplings among the spins. Although the system Hamiltonian is never fully specified, we demonstrate high fidelity control over the logical degrees of freedom. In fact, the DFS enc…
A new approach to efficient quantum computation with probabilistic gates is proposed and analyzed in both a local and non-local setting. It combines heralded gates previously studied for atom or atom-like qubits with logical encoding from linear optical quantum computation in order to perform high fidelity quantum gates across a quantum network. The error-detecting properties of the heralded operations ensure high fidelity while the encoding makes it possible to correct for failed attempts such that deterministic and high-quality gates can be achieved. Importantly, this is robust to photon los…
Logical qubit encoding and quantum error correction (QEC) have been experimentally demonstrated in various physical systems with multiple physical qubits, however, logical operations are challenging due to the necessary nonlocal operations. Alternatively, logical qubits with bosonic-mode-encoding are of particular interest because their QEC protection is hardware efficient, but gate operations on QEC protected logical qubits remain elusive. Here, we experimentally demonstrate full control on a single logical qubit with a binomial bosonic code, including encoding, decoding, repetitive QEC, and …
Fault-tolerant quantum error correction (QEC) is crucial for unlocking the true power of quantum computers. QEC codes use multiple physical qubits to encode a logical qubit, which is protected against errors at the physical qubit level. Here we use a trapped ion system to experimentally prepare $m$-qubit GHZ states and sample the measurement results to construct $m\times m$ logical states of the $[[m^2,1,m]]$ Shor code, up to $m=7$. The synthetic logical fidelity shows how deeper encoding can compensate for additional gate errors in state preparation for larger logical states. However, the opt…
We present a formalism for encoding the logical basis of a qubit into subspaces of multiple physical levels. The need for this multilevel encoding arises naturally in situations where the speed of quantum operations exceeds the limits imposed by the addressability of individual energy levels of the qubit physical system. A basic feature of the multilevel encoding formalism is the logical equivalence of different physical states and correspondingly, of different physical transformations. This logical equivalence is a source of a significant flexibility in designing logical operations, while the…
Encoding logical quantum information in harmonic oscillator modes is a promising and hardware-efficient approach to the realization of a quantum computer. In this work, we propose to encode logical qubits in grid states of an ensemble of harmonic oscillator modes. We first discuss general results about these multimode bosonic codes; how to design them, how to practically implement them in different experimental platforms and how lattice symmetries can be leveraged to perform logical non-Clifford operations. We then introduce in detail two two-mode grid codes based on the hypercubic and D4 latt…
We propose two protocols to encode a logical qubit into physical qubits relying on common types of qubit-qubit interactions in as simple forms as possible. We comment on its experimental implementation in several quantum computing architectures, e.g. with trapped atomic ion qubits, atomic qubits inside a high Q optical cavity, solid state Josephson junction qubits, and Bose-Einstein condensed atoms.
Geometric quantum computation offers a practical strategy toward robust quantum computation due to its inherently error tolerance. However, the rigorous geometric conditions lead to complex and/or error-disturbed quantum controls, especially for logical qubits that involve more physical qubits, whose error tolerance is effective in principle though, their experimental demonstration is still demanding. Thus, how to best simplify the needed control and manifest its full advantage has become the key to widespread applications of geometric quantum computation. Here we propose a new fast and robust…
A critical step in experimental quantum information processing (QIP) is to implement control of quantum systems protected against decoherence via informational encodings, such as quantum error correcting codes, noiseless subsystems and decoherence free subspaces. These encodings lead to the promise of fault tolerant QIP, but they come at the expense of resource overheads. Part of the challenge in studying control over multiple logical qubits, is that QIP test-beds have not had sufficient resources to analyze encodings beyond the simplest ones. The most relevant resources are the number of avai…
Practical quantum computing requires robust encoding of logical qubits in physical systems to protect fragile quantum information. Currently, the lack of scalability limits the logical encoding in most physical systems, and thus the high scalability of propagating light can be a game changer for realizing a practical quantum computer. However, propagating light also has a drawback: the difficulty of logical encoding due to weak nonlinearity. Here, we propose Gaussian breeding that encodes arbitrary Gottesman-Kitaev-Preskill (GKP) qubits in propagating light. The key idea is the efficient and i…
When the environmental disturbace to a quantum system has a wavelength much larger than the system size, all qubits localized within a small area are under action of the same error operators. Noiseless subsystem and decoherence free subspace are known to correct such collective errors. We construct simple quantum circuits, which implement these collective error correction codes, for a small number $n$ of physical qubits. A single logical qubit is encoded with $n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive relations among the subspaces employed in noiseless subs…
iSwitch: QEC on Demand via In-Situ Encoding of Bare Qubits for Ion Trap Architectures
University of California, San Diego（加州大学圣地亚哥分校）；Quantinuum；Cisco Quantum Lab（思科量子实验室）；University of California, Berkeley（加州大学伯克利分校）；University of California, Los Angeles（加州大学洛杉矶分校）；Oak Ridge National Laboratory（橡树岭国家实验室）
I calculate the fidelity of a [7,1,3] CSS quantum error correction code logical zero state constructed in a non-equiprobable Pauli operator error environment for two methods of encoding. The first method is to apply fault tolerant error correction to an arbitrary state of 7 qubits utilizing Shor states for syndrome measurement. The Shor states are themselves constructed in the non-equiprobable Pauli operator error environment and their fidelity depends on the number of verifications done to ensure multiple errors will not propagate into the encoded quantum information. Surprisingly, performing…
Quantum information stored in a qubit is rapidly lost to the environment. The realization of robust qubits is one of the most important challenges in quantum computing. Herein we propose to embed a logical qubit within the manifold of a qudit as a scheme to preserve quantum information over extended periods of time. Under identical conditions (e.g. decoherence channels) the submanifold of the logical qubit exhibits extended lifetimes compared to a pure two-level system (qubit). The retention of quantum information further improves with separation between the sublevels of the logical qubit. Lif…
We propose a scheme for encoding logical qubits in a subspace protected against collective rotations around the propagation axis using the polarization and transverse spatial degrees of freedom of single photons. This encoding allows for quantum key distribution without the need of a shared reference frame. We present methods to generate entangled states of two logical qubits using present day down-conversion sources and linear optics, and show that the application of these entangled logical states to quantum information schemes allows for alignment-free tests of Bell's inequalities, quantum d…
Stabilization of encoded logical qubits using quantum error correction is key to the realization of reliable quantum computers. While qubit codes require many physical systems to be controlled, oscillator codes offer the possibility to perform error correction on a single physical entity. One powerful encoding for oscillators is the grid state or GKP encoding, which allows small displacement errors to be corrected. Here we introduce and implement a dissipative map designed for physically realistic finite GKP codes which performs quantum error correction of a logical qubit implemented in the mo…
One of the main challenges for the manipulation and storage of multipartite entanglement is its fragility under noise. We present a simple recipe for the systematic enhancement of the resistance of multipartite entanglement against any local noise with a privileged direction in the Bloch sphere. For the case of exact local dephasing along any given basis, and for all noise strengths, our prescription grants full robustness: Even states with exponentially decaying entanglement are mapped to states whose entanglement is constant. In contrast to previous techniques resorting to complex logical-qu…
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 …
We show how to carry out quantum logical operations (controlled-not and Toffoli gates) on encoded qubits for several encodings which protect against various 1-bit errors. This improves the reliability of these operations by allowing one to correct for one bit errors which either preexisted or occurred in course of operation. The logical operations we consider allow one to cary out the vast majority of the steps in the quantum factoring algorithm. Thus, our results help bring quantum factoring and other quantum computations closer to reality
Fault-Tolerant Resource Comparison of Qudit and Qubit Encodings for Diagonal Quadratic Operators
University of Liverpool（利物浦大学）；Fermi National Accelerator Laboratory（费米国家加速器实验室）；University of Trento（特伦托大学）；INFN-TIFPA Trento Institute of Fundamental Physics and Applications（意大利国家核物理研究所-特伦托基础物理与应用研究所）
Doubling Qubits in a Trapped-Ion System via Vibrational Dual-Rail Encoding
State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University（清华大学物理系低维量子物理国家重点实验室）；Korea Institute for Advanced Study（韩国高等科学研究院）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）；Department of Chemistry, Yonsei University（延世大学化学系）
An outstanding challenge for quantum information processing using bosonic systems is Gaussian errors such as excitation loss and added thermal noise errors. Thus, bosonic quantum error correction (QEC) is essential. Most bosonic QEC schemes encode a finite-dimensional logical qubit or qudit into noisy bosonic oscillator modes. In this case, however, the infinite-dimensional bosonic nature of the physical system is lost at the error-corrected logical level. On the other hand, there are several proposals for encoding an oscillator mode into many noisy oscillator modes. However, these oscillator-…
Recent development of mixed-state encoding (MSE) allows pure-state logical information to be encoded by a bosonic (continuous-variable) system in mixed physical state. Despite interest due to its counter-intuitiveness, the utility of the current MSE scheme is limited due to several operational drawbacks, namely redundant information carrier, probabilistic initialisation, and requirement of discrete-variable measurement. In this work, we present a simplified MSE that does not suffer from any of these drawbacks. Specifically, our protocol encodes each qubit by only one mixed-state bosonic mode, …
Quantum computers will eventually reach a size at which quantum error correction becomes imperative. Quantum information can be protected from qubit imperfections and flawed control operations by encoding a single logical qubit in multiple physical qubits. This redundancy allows the extraction of error syndromes and the subsequent detection or correction of errors without destroying the logical state itself through direct measurement. Here we show the encoding and syndrome measurement of a fault-tolerant logical qubit via an error detection protocol on four physical qubits, represented by trap…
Fault-tolerant logical operations for qubits encoded by CSS codes are discussed, with emphasis on methods that apply to codes of high rate, encoding k qubits per block with k>1. It is shown that the logical qubits within a given block can be prepared by a single recovery operation in any state whose stabilizer generator separates into X and Z parts. Optimized methods to move logical qubits around and to achieve controlled-not and Toffoli gates are discussed. It is found that the number of time-steps required to complete a fault-tolerant quantum computation is the same when k>1 as when k=1.
Quantum error correcting codes protect quantum computation from errors caused by decoherence and other noise. Here we study the problem of designing logical operations for quantum error correcting codes. We present an automated procedure which generates logical operations given known encoding and correcting procedures. Our technique is to use variational circuits for learning both the logical gates and the physical operations implementing them. This procedure can be implemented on near-term quantum computers via quantum process tomography. It enables automatic discovery of logical gates from a…
Quantum error correction procedures have the potential to enable faithful operation of large-scale quantum computers. They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings. To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed. In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fai…
The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction ci…
