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## Quantum Error Correction Codes

## Quantum Error Correction For Beginners

## Córcoles, **Easwar Magesan& Srikanth J. **

## Contents |

Since the shots producing the error syndromes are evenly distributed throughout the different unitary rotation pulses on the code qubits, the effect on the code state is to reduce the number The fundamental measurement observables that are rotated by elements of are constructed from the calibration states by first normalizing the shots for each of the code qubit channels to lie in If errors occur on the physical qubits independently at random with probability p per gate or timestep, the fault-tolerant protocol has probability of logical error for a single logical gate or The simplest method, due to Shor, is very general but also requires the most overhead and is frequently the most susceptible to noise. http://caribtechsxm.com/quantum-error/quantum-computation-quantum-error-correcting-codes-and-information-theory.php

The complete gate sequence in our error detection experiments is presented in e, where the dark boxes indicate refocus pulses during every two-qubit gate on the two qubits not involved on Phys. The code will be able to **correct bit flip (X)** errors as if it had a distance d1 and to correct phase (Z) errors as if it had a distance d2. This process is repeated 19,200 times to gather sufficient statistics. https://en.wikipedia.org/wiki/Quantum_error_correction

Since the ZZ and XX parities are encoded into Q2 and Q4, respectively, pure bit-flip errors are detected by Q2, whereas pure phase-flip errors are detected by Q4.Full size imageError propagation Steffen, Jay M. The general unitary Kraus operator is and the probabilities for the different syndromes are given by equation (6). By adding extra qubits and carefully encoding the quantum state we wish to protect, a quantum system can be insulated to great extent against errors.

The following circuit performs a π/8 rotation, given an ancilla state ∣ψπ/8⟩ = ∣0⟩ + exp(iπ/4)∣1⟩: Here P is the π/4 phase rotation diag(1, i), and X is the bit flip. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Nebendahl, D. 5 Qubit Code Stabilizer Codes and Quantum Error Correction PhD thesis California Institute of Technology (1997).12.Dennis, E., Kitaev, A., Landahl, A. & Preskill, J.

F. Quantum Error Correction For Beginners The protocol detects an arbitrary quantum error on an encoded two-qubit entangled state via quantum non-demolition parity measurements on another pair of error syndrome qubits. Following the error detection protocol of Fig. 1c, we acquire single-shot measurements of the syndrome qubits and correlate independent measurements around various axes of the code qubits for quantum state tomography27. https://quantiki.org/wiki/quantum-error-correction-and-fault-tolerance-0 et al.

More sophisticated techniques for fault-tolerant error correction involve less interaction with the data but at the cost of more complicated ancilla states. Quantum Error Correction Book L. As of late 2004, estimates for this threshold indicate that it could be as high as 1-3% [4], provided that there are sufficiently many qubits available. Then we interact the ancilla **with the encoded data qubits** using gates from our stock of transversal gates and perform a fault-tolerant measurement.

- The cat state contains as many qubits as the operator M to be measured, and we perform the controlled-X, -Y, or -Z operations transversally from the appropriate qubits of the cat
- Given a codeword of a particular [[n, 1]] QECC, we can take each physical qubit and again encode it using the same code, producing an [[n2, 1]] QECC.
- The syndrome qubit states {M2,M4}={0,+} (black), {M2,M4}={0,−} (green), {M2,M4}={1,+} (red) and {M2,M4}={1,−} (blue) indicate the magnitude and nature of the error ɛ.
- Although previous work14,24,25,26 implemented parity checks on linear arrangements of qubits, our experiment goes beyond into the other planar dimension.
- Second, the reconstructed conditional states have little to no weight in the single-qubit subspace.
- Taylor & Francis.
- The Josephson junctions, patterned via electron beam lithography, are made by double-angle deposition of Al (layer thicknesses of 35 and 85 nm) followed by a liftoff process.

Another useful representation is to map the single-qubit Pauli operators I, X, Y, Z to the finite field GF(4), which sets up a connection between stabilizer codes and a subset of Since we always prepare the code qubits in the codeword state at the beginning of the quantum process, when no error is applied to Q1, state tomography of Q1 and Q3 Quantum Error Correction Codes Nat. Stabilizer Codes And Quantum Error Correction. Quantum error correction also employs syndrome measurements.

The observed contrast between the different syndrome qubit state populations, near 0.6 in Fig. 3, is commensurate with a master-equation simulation that takes into account the measured coherence times of our navigate to this website The variance in the state fidelity is computed via a bootstrapping protocol described in ref. 29 and the physicality is the sum of the negative eigenvalues of the linear inversion estimate. By encoding both the XX and the ZZ stabilizers in the four-qubit lattice, we can protect a maximally entangled state of the two-code qubits against an arbitrary error.To demonstrate the SC Then C is an ((n,K)) (binary) quantum error-correcting code (QECC) correcting the set of errors E = {Ea} iff ∃R s.t. Steane Code

The assignment fidelities are given in Supplementary Table 1.State tomographyThe conditional states of the code qubits (Q1 and Q3) for the different error types (I, X, Y and Z) were reconstructed A.D.C. When p < pt = 1/C, the fault-tolerance helps, decreasing the logical error rate. More about the author Figure 7 shows the circuit schematic **at chip level,** including the design of the qubit capacitance and coupling lines.Figure 6: Experimental setup.Detailed wiring scheme for all room temperature control electronics and

Note that the following procedure can be used to measure (non-fault-tolerantly) the eigenvalue of any (possibly multi-qubit) Pauli operator M: Produce an ancilla qubit in the state ∣ + ⟩ = ∣0⟩ + ∣1⟩. 5-qubit Quantum Error Correction We implement two-qubit echo cross-resonance (ECR) gates22, ECR=ZX90–XI, which are primitives for constructing controlled-NOT (CNOT) operations. This allows us to establish the degree of addressability error23 present in our system.

Unlike classical bits that are subject to only digital bit-flip errors, quantum bits are susceptible to a much larger spectrum of errors, for which any complete quantum error-correcting code must account. Detecting bit-flip errors in a logical qubit using stabilizer measurements. We need to add some additional gate outside the Clifford group to allow universal quantum computation; a single gate will suffice, such as the single-qubit π/8 phase rotation diag(1, exp(iπ/4)). Quantum Code 7 Fung for experimental contributions.

Cerf and U. By using this site, you agree to the Terms of Use and Privacy Policy. Maas, E. click site Gambetta1& Jerry M.

Scalable and robust randomized benchmarking of quantum processes. designed the experiments. The simplest way is to store the information multiple times, and—if these copies are later found to disagree—just take a majority vote; e.g. Andersen, "Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code," Nature Photonics 4, 700 (2010), doi:10.1038/nphoton.2010.168 Bibliography[edit] Daniel Lidar and Todd Brun, ed. (2013). "Quantum Error Correction".

Here, we can clearly see that a majority of the resulting measurements are located in the lower left quadrant, and we will use the notation {M2,M4}={0,+}, with both syndromes signalling a Lett. 108, 070502 (2012).CASPubMedArticleDownload referencesAcknowledgementsWe thank M. For H2, we perform the same procedure, but each 1 is instead replaced by X. Your cache administrator is webmaster.

Note that if the calibrations were perfect, then are equal to ZI, IZ and ZZ.For each of the 36 different measurement settings, we bin each shot according to the measurement results Similarly for the phase-flip operation ,and for Since the state after the SWAP gate is (the qubits are ordered and Q1, Q3 are the code qubits), the error syndromes are given Choosing any representation xab of ρab in some operator basis allows us to writewhere M is a constant matrix whose entries depend only on the choice of operator basis. Suppose further that a noisy error corrupts the three-bit state so that one bit is equal to zero but the other two are equal to one.