Low-overhead constructions for the fault-tolerant Toffoli gate

Jones, Cody

doi:10.1103/physreva.87.022328

Cited by 169 publications

(223 citation statements)

References 25 publications

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“…There are many constructions for these controlled gates [45][46][47]. Here we anticipate that the cost of state preparation for the CISD state will be sub-dominant to the cost of the simulation.…”

Section: A Circuit For State Preparation Based On CI Ansatzmentioning

confidence: 99%

“…This means that we also need to consider the cost of implementing Λ N (X) gates. Although highly time-efficient constructions for the multiply controlled circuits can be made using the circuits of [46], they require a large number of qubits. In order to ensure that the space complexity of state preparation does not dominate the algorithm, we use the less time-efficient construction of Barenco et al [45] to compile the Λ N (X) gates.…”

mentioning

confidence: 99%

“…The gate Λ 1 (H) requires 2 T -gates [44], and the Toffoli gates can be implemented, up to an irrelevant phase, using 4 T -gates [46,47] and an ancilla qubit. The entire process requires at most N + 4 qubits, which is typically less memory than is required for the quantum simulation and eigenvalue estimation phases of the algorithm.…”

mentioning

confidence: 99%

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Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation

et al. 2015

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Although the simulation of quantum chemistry is one of the most anticipated applications of quantum computing, the scaling of known upper bounds on the complexity of these algorithms is daunting. Prior work has bounded errors due to Trotterization in terms of the norm of the error operator and analyzed scaling with respect to the number of spin orbitals. However, we find that these error bounds can be loose by up to sixteen orders of magnitude for some molecules. Furthermore, numerical results for small systems fail to reveal any clear correlation between ground state error and number of spin orbitals. We instead argue that chemical properties, such as the maximum nuclear charge in a molecule and the filling fraction of orbitals, can be decisive for determining the cost of a quantum simulation. Our analysis motivates several strategies to use classical processing to further reduce the required Trotter step size and to estimate the necessary number of steps, without requiring additional quantum resources. Finally, we demonstrate improved methods for state preparation techniques which are asymptotically superior to proposals in the simulation literature.

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Section: A Circuit For State Preparation Based On CI Ansatzmentioning

confidence: 99%

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confidence: 99%

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Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation

et al. 2015

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“…This can be done in a variety of ways and has become, now, highly optimised. [116][117][118][119][120] The original distillation protocol 71,121 proceeds by taking 15 approximate copies of |a〉: 9ã 1 〉;…”

Section: Quantum Information Processing With Mzmsmentioning

confidence: 99%

Majorana zero modes and topological quantum computation

2015

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We provide a current perspective on the rapidly developing field of Majorana zero modes (MZMs) in solid-state systems. We emphasise the theoretical prediction, experimental realisation and potential use of MZMs in future information processing devices through braiding-based topological quantum computation (TQC). Well-separated MZMs should manifest non-Abelian braiding statistics suitable for unitary gate operations for TQC. Recent experimental work, following earlier theoretical predictions, has shown specific signatures consistent with the existence of Majorana modes localised at the ends of semiconductor nanowires in the presence of superconducting proximity effect. We discuss the experimental findings and their theoretical analyses, and provide a perspective on the extent to which the observations indicate the existence of anyonic MZMs in solid-state systems. We also discuss fractional quantum Hall systems (the 5/2 state), which have been extensively studied in the context of non-Abelian anyons and TQC. We describe proposed schemes for carrying out braiding with MZMs as well as the necessary steps for implementing TQC. INTRODUCTIONTopological quantum computation 1,2 is an approach to faulttolerant quantum computation in which the unitary quantum gates result from the braiding of certain topological quantum objects called 'anyons'. Anyons braid non-trivially: two counterclockwise exchanges do not leave the state of the system invariant, unlike in the cases of bosons or fermions. Anyons can arise in two ways: as localised excitations of an interacting quantum Hamiltonian 3 or as defects in an ordered system.

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“…With our circuit, CCZ states for implementing a teleported CCZ gate are produced also after just one measurement of the appropriate non-Pauli operators and one syndrome measurement. This circuit is less resource intensive than the postselected procedure used in [17] that measures non-Pauli operators multiple times and also less intensive than decomposing a logical CCZ into four T gates [38] that are each magically injected (e.g., using Ref. [37]).…”

Section: Resource Comparison Against Magic Statesmentioning

confidence: 99%

Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes

2016

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It is an oft-cited fact that no quantum code can support a set of fault-tolerant logical gates that is both universal and transversal. This no-go theorem is generally responsible for the interest in alternative universality constructions including magic state distillation. Widely overlooked, however, is the possibility of nontransversal, yet still fault-tolerant, gates that work directly on small quantum codes. Here, we demonstrate precisely the existence of such gates. In particular, we show how the limits of nontransversality can be overcome by performing rounds of intermediate error correction to create logical gates on stabilizer codes that use no ancillas other than those required for syndrome measurement. Moreover, the logical gates we construct, the most prominent examples being Toffoli and controlled-controlled-Z, often complete universal gate sets on their codes. We detail such universal constructions for the smallest quantum codes, the 5-qubit and 7-qubit codes, and then proceed to generalize the approach. One remarkable result of this generalization is that any nondegenerate stabilizer code with a complete set of fault-tolerant single-qubit Clifford gates has a universal set of fault-tolerant gates. Another is the interaction of logical qubits across different stabilizer codes, which, for instance, implies a broadly applicable method of code switching.

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Low-overhead constructions for the fault-tolerant Toffoli gate

Cited by 169 publications

References 25 publications

Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation

Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation

Majorana zero modes and topological quantum computation

Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes

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