Microsoft Technology Licensing, LLC

Matthew Benjamin Hastings

Jeongwan Haah

David B Wecker

Michael Hartley Freedman

Parsa Hassan Bonderson

Zhenghan Wang

David Alexander Aasen

David Poulin

Krysta Marie Svore

Chetan Vasudeo Nayak

Roman Mykolayovych Lutchyn

Torsten Karzig

Guang Hao Low

Robin Ashok Kothari

Aspects of the disclosure include removing a faulty qubit in a quantum circuit. The faulty qubit is determined to be in the quantum circuit, the faulty qubit being associated with a plaquette having other qubits, where adjacent plaquettes are neighboring the plaquette. A route is determined to isolate the plaquette from the adjacent plaquettes. Measurements are caused to be performed on the quantum circuit for the route that isolates the plaquette having the faulty qubit and the other qubits.

Fault tolerant Hastings-Haah codes in the presence of dead qubits

A method and apparatus for performing quantum error correction using an automorphism code related to the honeycomb code. The automorphism code is based on Kramers-Wannier (KW) duality. For embodiments on a hexagonal lattice using three repeated time steps, ⅓ of the pixels are active in a given time steps. In a given time step r, a KW circuit is applied to plaquettes labeled r mod 3, transferring quantum information from the active qubits at the beginning to a new set of active qubits at the end of the time step. Each of the three plaquette types is associated with one stabilizer either given by a product of six Z operators or three X operators. The KW circuit maps the product of X operators to the product of Z operators and vice-versa. The stabilizer group of the superlattice toric code changes every round.

Quantum code with simpler pairwise checks

A quantum error correcting code with dynamically generated logical qubits is provided. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, the measurement patterns generate logical qubits, allowing the code to act as a fault-tolerant quantum memory. Each measurement can be a two-qubit Pauli measurement.

Quantum code with pairwise checks

Boundary conditions for the honeycomb code

Disclosed herein are example quantum algorithms to solve certain problems (e.g., exactly) in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D; this problem is called weighted MAX-ED-LIN2. In some cases, it is desirable that the optimal solution be unique for odd D and doubly degenerate for even D; however, example algorithms still work without this condition and it is shown how to reduce to the case without this assumption at the cost of an additional overhead.

Short path quantum procedures for solving combinatorial optimization problems

Classical and quantum computational systems and methods for principal component analysis of multi-dimensional datasets are presented. A dataset is encoded in a tensor of rank p, where p is a positive integer that may be greater than 2. The classical methods are based on linear algebra. The quantum methods achieve a quartic speedup while using exponentially smaller space than the fastest classical algorithm, and a super-polynomial speedup over classical algorithms that use only polynomial space. In particular, an improved threshold for recovery is achieved. The presented classical and quantum methods work for both even and odd ranked tensors. Accordingly, quantum computation may be applied to large-scale inference problems, e.g., machine learning applications or other applications that involve highly-dimensional datasets.

Classical and quantum computation for principal component analysis of multi-dimensional datasets

In this disclosure, example quantum algorithms for approximate optimization based on a sudden quench of a Hamiltonian. While the algorithm is general, it is analyzed in this disclosure in the specific context of MAX-EK-LIN2, for both even and odd K. It is to be understood, however, that the algorithm can be generalized to other contexts. A duality can be found: roughly, either the algorithm provides some nontrivial improvement over random or there exist many solutions which are significantly worse than random. A classical approximation algorithm is then analyzed and a similar duality is found, though the quantum algorithm provides additional guarantees in certain cases.

Quantum procedures for approximate optimization by quenching

Examples are disclosed that relate to, on a quantum computing device, distilling magic states encoded in a [[n,k,d]] block code comprising an outer code. One example provides a method comprising preparing encoded noisy magic states using data qubits, and measuring Clifford stabilizers on the data qubits, thereby applying an inner code. The method further comprises initializing output qubits and initiating a teleportation of distilled magic states derived from the encoded noisy magic states to the output qubits. The method further comprises measuring X-stabilizers on the data qubits, postselecting based on the outcomes, measuring each data qubit destructively utilizing Z-stabilizers, and applying one or more postselection conditions to the data qubits to complete the teleportation of the distilled magic states to the output qubits.

Magic state distillation using inner and outer error correcting codes

Embodiments of the disclosed technology concern a quantum circuit configured to implement a real time evolution unitary of a Hamiltonian in a quantum computing device, wherein a unit time evolution unitary operator is decomposed into overlapping smaller blocks of unitary operators. In some implementations, (a) the size of the overlap is proportional to the logarithm of a number of qubits in the simulated system, (b) the size of the overlap is proportional to the logarithm of a total simulated evolution time, and/or (c) the size of the overlap is proportional to a Lieb-Robinson velocity.

Quantum simulation of real time evolution of lattice Hamiltonians

Disclosed herein are example embodiments of protocols to distill magic states for T-gates. Particular examples have low space overhead and use an asymptotically optimal number of input magic states to achieve a given target error. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T-gate depth of the circuit scale linearly in the logarithm of the target error. Unlike other distillation protocols, examples of the disclosed protocol achieve this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. Embodiments of the protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. Embodiments of the protocol rely on the construction of weakly self-dual Calderbank-Shor-Steane codes ('CSS codes') with many logical qubits and large distance, allowing one to implement control-Swaps on multiple qubits. This code is referred to herein as the 'inner code'. The control-Swaps are then used to measure properties of the magic state and detect errors, using another code that is referred to as the &#x201c;outer code&#x201d;. Alternatively, one can use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. Several specific small examples of this protocol are disclosed herein.

Magic state distillation with low space overhead and asymptotic input count

This application concerns quantum computing and quantum circuits. For example, among the embodiments disclosed herein are codes and protocols to distill T, controlled-S, and Toffoli (or CCZ) gates for use in croantum circuits. Examples of the disclosed codes use lower overhead for a given target accuracy relative to other distillation techniques. In some embodiments, a magic state distillation protocol is generated for creating magic states in the quantum computing device, wherein the magic state distillation protocol includes (a) Reed-Muller codes, or (b) punctured Reed-Muller codes. The quantum computing device can then configured to implement the magic state distillation protocol.

Codes and protocols for distilling T, controlled-S, and toffoli gates

The disclosed technology concerns tools and techniques for implementing error-correction codes in a quantum computing device. In particular embodiments, Majorana fermion stabilizer codes having small numbers of modes and distance are disclosed. Particular embodiments have an upper bound on the number of logical qubits for distance 4 codes, and Majorana fermion codes are constructed that saturate this bound. Other distance 4 and 6 codes are also disclosed.

Small Majorana fermion codes

Quantum computations based on second quantization are performed by applying one body and two body terms in a selected order. Typically, terms associated with operators that commute are applied prior to application of other terms. In a particular example, one body terms of the form hare applied first, followed by two body terms of the form h.

Classical simulation constants and ordering for quantum chemistry simulation

Disclosed herein are example embodiments of protocols to distill magic states for T-gates. Particular examples have low space overhead and use an asymptotically optimal number of input magic states to achieve a given target error. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T-gate depth of the circuit scale linearly in the logarithm of the target error. Unlike other distillation protocols, examples of the disclosed protocol achieve this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. Embodiments of the protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. Embodiments of the protocol rely on the construction of weakly self-dual Calderbank-Shor-Steane codes ('CSS codes') with many logical qubits and large distance, allowing one to implement control-Swaps on multiple qubits. This code is referred to herein as the 'inner code'. The control-Swaps are then used to measure properties of the magic state and detect errors, using another code that is referred to as the 'outer code'. Alternatively, one can use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. Several specific small examples of this protocol are disclosed herein.

Among the embodiments disclosed herein are example methods for generating all Clifford gates for a system of Majorana Tetron qubits (quasiparticle poisoning protected) given the ability to perform certain 4 Majorana zero mode measurements. Also disclosed herein are example designs for scalable quantum computing architectures that enable the methods for generating the Clifford gates, as well as other operations on the states of MZMs. These designs are configured in such a way as to allow the generation of all the Clifford gates with topological protection and non-Clifford gates (e.g. a &#x3c0;/8-phase gate) without topological protection, thereby producing a computationally universal gate set. Several possible realizations of these architectures are disclosed.

Growing community of inventors

Seattle, WA, United States of America

Matthew Benjamin Hastings