Accuracy threshold for postselected quantum computation

"... We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the ..."

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We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary k-qubit gates each of whose input wires is subject to depolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for p> 1 − Θ(1 / √ k), the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of k = 2, our bound is p> 35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes 29.3%. These bounds on p are notably better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, which we believe should lead to further progress in deriving such bounds. 1

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Quantum universality by state distillation

by Ben W. Reichardt , 2009

"... Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This “magic states distillation ” question is closely related to quantum fault tolerance. Lower bounds on the noi ..."

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Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This “magic states distillation ” question is closely related to quantum fault tolerance. Lower bounds on the noise tolerable on the ancilla help give lower bounds on the tolerable noise rate threshold for fault-tolerant computation. Upper bounds show the limits of threshold upperbound arguments based on the Gottesman-Knill theorem. We extend the range of single-qubit mixed states that are known to give universality, by using a simple parity-checking operation. For applications to proving threshold lower bounds, certain practical stability characteristics are often required, and we also show a stable distillation procedure. No distillation upper bounds are known beyond those given by the Gottesman-Knill theorem. One might ask whether distillation upper bounds reduce to upper bounds for single-qubit ancilla states. For multi-qubit pure states and previously considered two-qubit ancilla states, the answer is yes. However, we exhibit two-qubit mixed states that are not mixtures of stabilizer states, but for which every postselected stabilizer reduction from two qubits to one outputs a mixture of stabilizer states. Distilling such states would require true multi-qubit state distillation methods. 1

Quantum Computers: Noise Propagation and Adversarial Noise Models

by Gil Kalai , 2009

"... In this paper we consider adversarial noise models that will fail quantum error correction and fault-tolerant quantum computation. We describe known results regarding high-rate noise, sequential computation, and reversible noisy computation. We continue by discussing highly correlated noise and the ..."

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In this paper we consider adversarial noise models that will fail quantum error correction and fault-tolerant quantum computation. We describe known results regarding high-rate noise, sequential computation, and reversible noisy computation. We continue by discussing highly correlated noise and the “boundary, ” in terms of correlation of errors, of the “threshold theorem. ” Next, we draw a picture of adversarial forms of noise called (collectively) “detrimental noise.” Detrimental noise is modeled after familiar properties of noise propagation. However, it can have various causes. We start by pointing out the difference between detrimental noise and standard noise models for two qubits and proceed to a discussion of highly entangled states, the rate of noise, and general noisy quantum systems. Research supported in part by an NSF grant, an ISF grant, and a BSF grant.

Quantum Error Correction Code in the Hamiltonian Formulation

by Yong Zhang , 2008

"... The Hamiltonian model of quantum error correction code in the literature is often constructed with the help of its stabilizer formalism. But there have been many known examples of nonadditive codes which are beyond the standard quantum error correction theory using the stabilizer formalism. In this ..."

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The Hamiltonian model of quantum error correction code in the literature is often constructed with the help of its stabilizer formalism. But there have been many known examples of nonadditive codes which are beyond the standard quantum error correction theory using the stabilizer formalism. In this paper, we suggest the other type of Hamiltonian formalism for quantum error correction code without involving the stabilizer formalism, and explain it by studying the Shor nine-qubit code and its generalization. In this Hamiltonian formulation, the unitary evolution operator at a specific time is a unitary basis transformation matrix from the product basis to the quantum error correction code. This basis transformation matrix acts as an entangling quantum operator transforming a separate state to an entangled one, and hence the entanglement nature of the quantum error correction code can be explicitly shown up. Furthermore, as it forms a unitary representation of the Artin braid group, the quantum error correction code can be described by a braiding operator. Moreover, as the unitary evolution operator is a solution of the quantum Yang–Baxter equation, the corresponding Hamiltonian model can be explained as an integrable model in the Yang–Baxter theory. On the other hand, we generalize the Shor nine-qubit code and articulate a topic called quantum error correction codes using Greenberger-Horne-Zeilinger states to yield new nonadditive codes and channel-adapted codes.

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Fault-Tolerant Quantum Computation with Constant Overhead

by Daniel Gottesman

"... ar ..."

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by Ching-yi Lai, Gerardo Paz, Martin Suchara, Todd A. Brun

"... Knill demonstrated a fault-tolerant quantum computation scheme based on concatenated error-detecting codes and postselection with a simulated error threshold of 3 % over the depolarizing channel. We show how to use Knill’s postselection scheme in a practical two-dimensional quantum architecture that ..."

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Knill demonstrated a fault-tolerant quantum computation scheme based on concatenated error-detecting codes and postselection with a simulated error threshold of 3 % over the depolarizing channel. We show how to use Knill’s postselection scheme in a practical two-dimensional quantum architecture that we designed with the goal to optimize the error correction properties, while satisfying important architectural constraints. In our 2D architecture, one logical qubit is embedded in a tile consisting of 5×5 physical qubits. The movement of these qubits is modeled as noisy SWAP gates and the only physical operations that are allowed are local one- and two-qubit gates. We evaluate the practical properties of our design, such as its error threshold, and compare it to the concatenated Bacon-Shor code and the concatenated Steane code. Assuming that all gates have the same error rates, we obtain a threshold of 3.06 × 10−4 in a local adversarial stochastic noise model, which is the highest known error threshold for concatenated codes in 2D. We also present a Monte Carlo simulation of the 2D architecture with depolarizing noise and we calculate a pseudo-threshold of about 0.1%. With memory error rates one-tenth of the worst gate error rates, the threshold for the adversarial noise model, and the pseudo-threshold over depolarizing noise, are 4.06 × 10−4 and 0.2%, respectively. In a hypothetical technology where memory error rates are negligible, these thresholds can be further increased by shrinking the tiles into a 4 × 4 layout.

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by Sergey Bravyi, David Divincenzo, Daniel Loss , 2008

"... Polynomial-time algorithm for simulation of weakly interacting quantum spin systems ..."

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Polynomial-time algorithm for simulation of weakly interacting quantum spin systems

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SURVEY ON THE BOUNDS OF THE QUANTUM FAULT-TOLERANCE THRESHOLD

by Christopher Graves

"... I rst brie y summarize the threshold theorem and describe the motivations for tightening the bounds on the threshold quantum decoherence rate. I then go on to summarize and organize recent results regarding both the lower and the upper ..."

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I rst brie y summarize the threshold theorem and describe the motivations for tightening the bounds on the threshold quantum decoherence rate. I then go on to summarize and organize recent results regarding both the lower and the upper