16 research outputs found
Three-dimensional surface codes: Transversal gates and fault-tolerant architectures
One of the leading quantum computing architectures is based on the
two-dimensional (2D) surface code. This code has many advantageous properties
such as a high error threshold and a planar layout of physical qubits where
each physical qubit need only interact with its nearest neighbours. However,
the transversal logical gates available in 2D surface codes are limited. This
means that an additional (resource intensive) procedure known as magic state
distillation is required to do universal quantum computing with 2D surface
codes. Here, we examine three-dimensional (3D) surface codes in the context of
quantum computation. We introduce a picture for visualizing 3D surface codes
which is useful for analysing stacks of three 3D surface codes. We use this
picture to prove that the and gates are transversal in 3D surface
codes. We also generalize the techniques of 2D surface code lattice surgery to
3D surface codes. We combine these results and propose two quantum computing
architectures based on 3D surface codes. Magic state distillation is not
required in either of our architectures. Finally, we show that a stack of three
3D surface codes can be transformed into a single 3D color code (another type
of quantum error-correcting code) using code concatenation.Comment: 23 pages, 24 figures, v2: published versio
Partitioning qubits in hypergraph product codes to implement logical gates
The promise of high-rate low-density parity check (LDPC) codes to substantially reduce the overhead of fault-tolerant quantum computation depends on constructing efficient, fault-tolerant implementations of logical gates on such codes. Transversal gates are the simplest type of fault-tolerant gate, but the potential of transversal gates on LDPC codes has hitherto been largely neglected. We investigate the transversal gates that can be implemented in hypergraph product codes, a class of LDPC codes. Our analysis is aided by the construction of a symplectic canonical basis for the logical operators of hypergraph product codes, a result that may be of independent interest. We show that in these codes transversal gates can implement Hadamard (up to logical SWAP gates) and control-Z on all logical qubits. Moreover, we show that sequences of transversal operations, interleaved with error correction, allow implementation of entangling gates between arbitrary pairs of logical qubits in the same code block. We thereby demonstrate that transversal gates can be used as the basis for universal quantum computing on LDPC codes, when supplemented with state injection
Cellular automaton decoders for topological quantum codes with noisy measurements and beyond
We propose an error correction procedure based on a cellular automaton, the
sweep rule, which is applicable to a broad range of codes beyond topological
quantum codes. For simplicity, however, we focus on the three-dimensional (3D)
toric code on the rhombic dodecahedral lattice with boundaries and prove that
the resulting local decoder has a non-zero error threshold. We also numerically
benchmark the performance of the decoder in the setting with measurement errors
using various noise models. We find that this error correction procedure is
remarkably robust against measurement errors and is also essentially
insensitive to the details of the lattice and noise model. Our work constitutes
a step towards finding simple and high-performance decoding strategies for a
wide range of quantum low-density parity-check codes.Comment: 16 pages, 10 figures, v2: published versio
Implementing fault-tolerant non-Clifford gates using the [[8,3,2]] color code
Quantum computers promise to solve problems that are intractable for
classical computers, but qubits are vulnerable to many sources of error,
limiting the depth of the circuits that can be reliably executed on today's
quantum hardware. Quantum error correction has been proposed as a solution to
this problem, whereby quantum information is protected by encoding it into a
quantum error-correcting code. But protecting quantum information is not
enough, we must also process the information using logic gates that are robust
to faults that occur during their execution. One method for processing
information fault-tolerantly is to use quantum error-correcting codes that have
logical gates with a tensor product structure (transversal gates), making them
naturally fault-tolerant. Here, we test the performance of a code with such
transversal gates, the [[8,3,2]] color code, using trapped-ion and
superconducting hardware. We observe improved performance (compared to no
encoding) for encoded circuits implementing non-Clifford gates, a class of
gates that are essential for achieving universal quantum computing. In
particular, we find improved performance for an encoded circuit implementing
the control-control gate, a key gate in Shor's algorithm. Our results
illustrate the potential of using codes with transversal gates to implement
non-trivial algorithms on near-term quantum hardware.Comment: 7+5 pages, comments welcom
Machine learning logical gates for quantum error correction
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
analytically designed error correcting codes and can be extended to error
correcting codes found by numerical optimizations. We test the procedure by
simulation on classical computers on small quantum codes of four qubits to
fifteen qubits and show that it finds most logical gates known in the current
literature. Additionally, it generates logical gates not found in the current
literature for the [[5,1,2]] code, the [[6,3,2]] code, and the [[8,3,2]] code.Comment: 17 page
Tailoring three-dimensional topological codes for biased noise
Tailored topological stabilizer codes in two dimensions have been shown to
exhibit high storage threshold error rates and improved subthreshold
performance under biased Pauli noise. Three-dimensional (3D) topological codes
can allow for several advantages including a transversal implementation of
non-Clifford logical gates, single-shot decoding strategies, parallelized
decoding in the case of fracton codes as well as construction of fractal
lattice codes. Motivated by this, we tailor 3D topological codes for enhanced
storage performance under biased Pauli noise. We present Clifford deformations
of various 3D topological codes, such that they exhibit a threshold error rate
of under infinitely biased Pauli noise. Our examples include the 3D
surface code on the cubic lattice, the 3D surface code on a checkerboard
lattice that lends itself to a subsystem code with a single-shot decoder, the
3D color code, as well as fracton models such as the X-cube model, the
Sierpinski model and the Haah code. We use the belief propagation with ordered
statistics decoder (BP-OSD) to study threshold error rates at finite bias. We
also present a rotated layout for the 3D surface code, which uses roughly half
the number of physical qubits for the same code distance under appropriate
boundary conditions. Imposing coprime periodic dimensions on this rotated
layout leads to logical operators of weight at infinite bias and a
corresponding subthreshold scaling of the logical failure rate,
where is the number of physical qubits in the code. Even though this
scaling is unstable due to the existence of logical representations with
low-rate Pauli errors, the number of such representations scales only
polynomially for the Clifford-deformed code, leading to an enhanced effective
distance.Comment: 51 pages, 34 figure
Graphical CSS Code Transformation Using ZX Calculus
In this work, we present a generic approach to transform CSS codes by
building upon their equivalence to phase-free ZX diagrams. Using the ZX
calculus, we demonstrate diagrammatic transformations between encoding maps
associated with different codes. As a motivating example, we give explicit
transformations between the Steane code and the quantum Reed-Muller code, since
by switching between these two codes, one can obtain a fault-tolerant universal
gate set. To this end, we propose a bidirectional rewrite rule to find a (not
necessarily transversal) physical implementation for any logical ZX diagram in
any CSS code.
Then we focus on two code transformation techniques: code morphing, a
procedure that transforms a code while retaining its fault-tolerant gates, and
gauge fixing, where complimentary codes can be obtained from a common subsystem
code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]]
code). We provide explicit graphical derivations for these techniques and show
how ZX and graphical encoder maps relate several equivalent perspectives on
these code-transforming operations.Comment: In Proceedings QPL 2023, arXiv:2308.1548
Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer
Photonics is the platform of choice to build a modular, easy-to-network
quantum computer operating at room temperature. However, no concrete
architecture has been presented so far that exploits both the advantages of
qubits encoded into states of light and the modern tools for their generation.
Here we propose such a design for a scalable and fault-tolerant photonic
quantum computer informed by the latest developments in theory and technology.
Central to our architecture is the generation and manipulation of
three-dimensional hybrid resource states comprising both bosonic qubits and
squeezed vacuum states. The proposal enables exploiting state-of-the-art
procedures for the non-deterministic generation of bosonic qubits combined with
the strengths of continuous-variable quantum computation, namely the
implementation of Clifford gates using easy-to-generate squeezed states.
Moreover, the architecture is based on two-dimensional integrated photonic
chips used to produce a qubit cluster state in one temporal and two spatial
dimensions. By reducing the experimental challenges as compared to existing
architectures and by enabling room-temperature quantum computation, our design
opens the door to scalable fabrication and operation, which may allow photonics
to leap-frog other platforms on the path to a quantum computer with millions of
qubits.Comment: 38 pages, many figures. Comments welcom
Fault-tolerant quantum computing with three-dimensional surface codes
Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits