Tag Archives: Quantum Computing

The Question of Quantum Supremacy

Posted by Sergio Boixo, Research Scientist and Theory Team Lead, and Charles Neill, Quantum Electronics Engineer, Quantum A.I. Lab

Quantum computing integrates the two largest technological revolutions of the last half century, information technology and quantum mechanics. If we compute using the rules of quantum mechanics, instead of binary logic, some intractable computational tasks become feasible. An important goal in the pursuit of a universal quantum computer is the determination of the smallest computational task that is prohibitively hard for today’s classical computers. This crossover point is known as the “quantum supremacy” frontier, and is a critical step on the path to more powerful and useful computations.

In “Characterizing quantum supremacy in near-term devices” published in Nature Physics (arXiv here), we present the theoretical foundation for a practical demonstration of quantum supremacy in near-term devices. It proposes the task of sampling bit-strings from the output of random quantum circuits, which can be thought of as the “hello world” program for quantum computers. The upshot of the argument is that the output of random chaotic systems (think butterfly effect) become very quickly harder to predict the longer they run. If one makes a random, chaotic qubit system and examines how long a classical system would take to emulate it, one gets a good measure of when a quantum computer could outperform a classical one. Arguably, this is the strongest theoretical proposal to prove an exponential separation between the computational power of classical and quantum computers.

Determining where exactly the quantum supremacy frontier lies for sampling random quantum circuits has rapidly become an exciting area of research. On one hand, improvements in classical algorithms to simulate quantum circuits aim to increase the size of the quantum circuits required to establish quantum supremacy. This forces an experimental quantum device with a sufficiently large number of qubits and low enough error rates to implement circuits of sufficient depth (i.e the number of layers of gates in the circuit) to achieve supremacy. On the other hand, we now understand better how the particular choice of the quantum gates used to build random quantum circuits affects the simulation cost, leading to improved benchmarks for near-term quantum supremacy (available for download here), which are in some cases quadratically more expensive to simulate classically than the original proposal.

Sampling from random quantum circuits is an excellent calibration benchmark for quantum computers, which we call cross-entropy benchmarking. A successful quantum supremacy experiment with random circuits would demonstrate the basic building blocks for a large-scale fault-tolerant quantum computer. Furthermore, quantum physics has not yet been tested for highly complex quantum states such as this.
Space-time volume of a quantum circuit computation. The computational cost for quantum simulation increases with the volume of the quantum circuit, and in general grows exponentially with the number of qubits and the circuit depth. For asymmetric grids of qubits, the computational space-time volume grows slower with depth than for symmetric grids, and can result in circuits exponentially easier to simulate.
In “A blueprint for demonstrating quantum supremacy with superconducting qubits” (arXiv here), we illustrate a blueprint towards quantum supremacy and experimentally demonstrate a proof-of-principle version for the first time. In the paper, we discuss two key ingredients for quantum supremacy: exponential complexity and accurate computations. We start by running algorithms on subsections of the device ranging from 5 to 9 qubits. We find that the classical simulation cost grows exponentially with the number of qubits. These results are intended to provide a clear example of the exponential power of these devices. Next, we use cross-entropy benchmarking to compare our results against that of an ordinary computer and show that our computations are highly accurate. In fact, the error rate is low enough to achieve quantum supremacy with a larger quantum processor.

Beyond achieving quantum supremacy, a quantum platform should offer clear applications. In our paper, we apply our algorithms towards computational problems in quantum statistical-mechanics using complex multi-qubit gates (as opposed to the two-qubit gates designed for a digital quantum processor with surface code error correction). We show that our devices can be used to study fundamental properties of materials, e.g. microscopic differences between metals and insulators. By extending these results to next-generation devices with ~50 qubits, we hope to answer scientific questions that are beyond the capabilities of any other computing platform.
Photograph of two gmon superconducting qubits and their tunable coupler developed by Charles Neill and Pedram Roushan.
These two publications introduce a realistic proposal for near-term quantum supremacy, and demonstrate a proof-of-principle version for the first time. We will continue to decrease the error rates and increase the number of qubits in quantum processors to reach the quantum supremacy frontier, and to develop quantum algorithms for useful near-term applications.

Source: Google AI Blog


Reformulating Chemistry for More Efficient Quantum Computation

Posted by Ryan Babbush, Senior Research Scientist, Quantum AI Team

The first known classical “computer” was the Antikythera mechanism, an analog machine used to simulate the classical mechanics governing dynamics of celestial bodies on an astronomical scale. Similarly, a major ambition of quantum computers is to simulate the quantum mechanics governing dynamics of particles on the atomic scale. These simulations are often classically intractable due to the complex quantum mechanics at play. Of particular interest is the simulation of electrons forming chemical bonds, which give rise to the properties of essentially all molecules, materials and chemical reactions.
Left: The first known computing device, the Antikythera mechanism: a classical machine used to simulate classical mechanics. Right: Google’s 22 Xmon qubit “foxtail” chip arranged in a bilinear array on a wafer, the predecessor to Google’s new Bristlecone quantum processor with 72 qubits, a quantum machine we intend to use to simulate quantum mechanics, among other applications.
Since the launch of the Quantum AI team in 2013, we have been developing practical algorithms for quantum processors. In 2015, we conducted the first quantum chemistry experiment on a superconducting quantum computing device, published in Physical Review X. More recently, our quantum simulation effort experimentally simulated exotic phases of matter and released the first software package for quantum computing chemistry, OpenFermion. Earlier this month, our hardware team announced the new Bristlecone quantum processor with 72 qubits.

Today, we highlight two recent publications with theoretical advances that significantly reduce the cost of these quantum computations. Our results were presented at the Quantum Information Processing and IBM ThinkQ conferences.

The first of these works, “Low-Depth Quantum Simulation of Materials,” published this week in Physical Review X, was a collaboration between researchers at Google, the group of Professor Garnet Chan at Caltech and the QuArC group at Microsoft. Our fundamental advance was to realize that by changing how molecules are represented on quantum computers, we can greatly simplify the quantum circuits required to solve the problem. Specifically, we specially design basis sets so that the equations describing the system energies (i.e. the Hamiltonian) become more straightforward to express for quantum computation.

To do this, we focused on using basis sets related to functions (plane waves) used in classical electronic structure calculations to provide a periodic representation of the physical system. This enables one to go beyond the quantum simulation of single-molecules and instead use quantum computers to model realistic materials. For instance, instead of simulating a single lithium hydride molecule floating in free space, with our approach one can quantum simulate a crystal of lithium hydride, which is how the material appears in nature. With larger quantum computers one could study other important materials problems such as the degradation of battery cathodes, chemical reactions involving heterogeneous catalysts, or the unusual electrical properties of graphene and superconductors.

In “Quantum Simulation of Electronic Structure with Linear Depth and Connectivity,” published last week in Physical Review Letters with the same collaborators and a Google intern from the Aspuru-Guzik group at Harvard, we leverage the structure introduced in the work above to design algorithms for near-term quantum computers with qubits laid out in a linear array. Whereas past methods required such quantum computers to run for time scaling as the fifth power of the number of simulated electrons for each dynamic step, our improved algorithm runs for time scaling linearly with respect to the number of electrons. This reduction in computational cost makes it viable to perform quantum chemistry simulations on near-term devices with fewer gates in each quantum circuit, possibly avoiding the need for full error-correction.

Even with these improvements, it is no small task to deploy such new technology to outperform classical quantum chemistry algorithms and methods which have been refined in parallel with the development of classical computers for more than eighty years. However, at the current rate of advances in quantum algorithms and hardware, quantum technologies may provide chemists with an invaluable new tool. We look forward to sharing our research results as they develop.

Reformulating Chemistry for More Efficient Quantum Computation

Posted by Ryan Babbush, Senior Research Scientist, Quantum AI Team

The first known classical “computer” was the Antikythera mechanism, an analog machine used to simulate the classical mechanics governing dynamics of celestial bodies on an astronomical scale. Similarly, a major ambition of quantum computers is to simulate the quantum mechanics governing dynamics of particles on the atomic scale. These simulations are often classically intractable due to the complex quantum mechanics at play. Of particular interest is the simulation of electrons forming chemical bonds, which give rise to the properties of essentially all molecules, materials and chemical reactions.
Left: The first known computing device, the Antikythera mechanism: a classical machine used to simulate classical mechanics. Right: Google’s 22 Xmon qubit “foxtail” chip arranged in a bilinear array on a wafer, the predecessor to Google’s new Bristlecone quantum processor with 72 qubits, a quantum machine we intend to use to simulate quantum mechanics, among other applications.
Since the launch of the Quantum AI team in 2013, we have been developing practical algorithms for quantum processors. In 2015, we conducted the first quantum chemistry experiment on a superconducting quantum computing device, published in Physical Review X. More recently, our quantum simulation effort experimentally simulated exotic phases of matter and released the first software package for quantum computing chemistry, OpenFermion. Earlier this month, our hardware team announced the new Bristlecone quantum processor with 72 qubits.

Today, we highlight two recent publications with theoretical advances that significantly reduce the cost of these quantum computations. Our results were presented at the Quantum Information Processing and IBM ThinkQ conferences.

The first of these works, “Low-Depth Quantum Simulation of Materials,” published this week in Physical Review X, was a collaboration between researchers at Google, the group of Professor Garnet Chan at Caltech and the QuArC group at Microsoft. Our fundamental advance was to realize that by changing how molecules are represented on quantum computers, we can greatly simplify the quantum circuits required to solve the problem. Specifically, we specially design basis sets so that the equations describing the system energies (i.e. the Hamiltonian) become more straightforward to express for quantum computation.

To do this, we focused on using basis sets related to functions (plane waves) used in classical electronic structure calculations to provide a periodic representation of the physical system. This enables one to go beyond the quantum simulation of single-molecules and instead use quantum computers to model realistic materials. For instance, instead of simulating a single lithium hydride molecule floating in free space, with our approach one can quantum simulate a crystal of lithium hydride, which is how the material appears in nature. With larger quantum computers one could study other important materials problems such as the degradation of battery cathodes, chemical reactions involving heterogeneous catalysts, or the unusual electrical properties of graphene and superconductors.

In “Quantum Simulation of Electronic Structure with Linear Depth and Connectivity,” published last week in Physical Review Letters with the same collaborators and a Google intern from the Aspuru-Guzik group at Harvard, we leverage the structure introduced in the work above to design algorithms for near-term quantum computers with qubits laid out in a linear array. Whereas past methods required such quantum computers to run for time scaling as the fifth power of the number of simulated electrons for each dynamic step, our improved algorithm runs for time scaling linearly with respect to the number of electrons. This reduction in computational cost makes it viable to perform quantum chemistry simulations on near-term devices with fewer gates in each quantum circuit, possibly avoiding the need for full error-correction.

Even with these improvements, it is no small task to deploy such new technology to outperform classical quantum chemistry algorithms and methods which have been refined in parallel with the development of classical computers for more than eighty years. However, at the current rate of advances in quantum algorithms and hardware, quantum technologies may provide chemists with an invaluable new tool. We look forward to sharing our research results as they develop.

Source: Google AI Blog


A Preview of Bristlecone, Google’s New Quantum Processor

Posted by Julian Kelly, Research Scientist, Quantum AI Lab

The goal of the Google Quantum AI lab is to build a quantum computer that can be used to solve real-world problems. Our strategy is to explore near-term applications using systems that are forward compatible to a large-scale universal error-corrected quantum computer. In order for a quantum processor to be able to run algorithms beyond the scope of classical simulations, it requires not only a large number of qubits. Crucially, the processor must also have low error rates on readout and logical operations, such as single and two-qubit gates.

Today we presented Bristlecone, our new quantum processor, at the annual American Physical Society meeting in Los Angeles. The purpose of this gate-based superconducting system is to provide a testbed for research into system error rates and scalability of our qubit technology, as well as applications in quantum simulation, optimization, and machine learning.
Bristlecone is Google’s newest quantum processor (left). On the right is a cartoon of the device: each “X” represents a qubit, with nearest neighbor connectivity.
The guiding design principle for this device is to preserve the underlying physics of our previous 9-qubit linear array technology1, 2, which demonstrated low error rates for readout (1%), single-qubit gates (0.1%) and most importantly two-qubit gates (0.6%) as our best result. This device uses the same scheme for coupling, control, and readout, but is scaled to a square array of 72 qubits. We chose a device of this size to be able to demonstrate quantum supremacy in the future, investigate first and second order error-correction using the surface code, and to facilitate quantum algorithm development on actual hardware.
2D conceptual chart showing the relationship between error rate and number of qubits. The intended research direction of the Quantum AI Lab is shown in red, where we hope to access near-term applications on the road to building an error corrected quantum computer.
Before investigating specific applications, it is important to quantify a quantum processor’s capabilities. Our theory team has developed a benchmarking tool for exactly this task. We can assign a single system error by applying random quantum circuits to the device and checking the sampled output distribution against a classical simulation. If a quantum processor can be operated with low enough error, it would be able to outperform a classical supercomputer on a well-defined computer science problem, an achievement known as quantum supremacy. These random circuits must be large in both number of qubits as well as computational length (depth). Although no one has achieved this goal yet, we calculate quantum supremacy can be comfortably demonstrated with 49 qubits, a circuit depth exceeding 40, and a two-qubit error below 0.5%. We believe the experimental demonstration of a quantum processor outperforming a supercomputer would be a watershed moment for our field, and remains one of our key objectives.
A Bristlecone chip being installed by Research Scientist Marissa Giustina at the Quantum AI Lab in Santa Barbara
We are looking to achieve similar performance to the best error rates of the 9-qubit device, but now across across all 72 qubits of Bristlecone. We believe Bristlecone would then be a compelling proof-of-principle for building larger scale quantum computers. Operating a device such as Bristlecone at low system error requires harmony between a full stack of technology ranging from software and control electronics to the processor itself. Getting this right requires careful systems engineering over several iterations.

We are cautiously optimistic that quantum supremacy can be achieved with Bristlecone, and feel that learning to build and operate devices at this level of performance is an exciting challenge! We look forward to sharing the results and allowing collaborators to run experiments in the future.

Announcing OpenFermion: The Open Source Chemistry Package for Quantum Computers

Crossposted on the Google Research Blog

“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.”
-Paul Dirac, Quantum Mechanics of Many-Electron Systems (1929)

In this passage, physicist Paul Dirac laments that while quantum mechanics accurately models all of chemistry, exactly simulating the associated equations appears intractably complicated. Not until 1982 would Richard Feynman suggest that instead of surrendering to the complexity of quantum mechanics, we might harness it as a computational resource. Hence, the original motivation for quantum computing: by operating a computer according to the laws of quantum mechanics, one could efficiently unravel exact simulations of nature. Such simulations could lead to breakthroughs in areas such as photovoltaics, batteries, new materials, pharmaceuticals and superconductivity. And while we do not yet have a quantum computer large enough to solve classically intractable problems in these areas, rapid progress is being made. Last year, Google published this paper detailing the first quantum computation of a molecule using a superconducting qubit quantum computer. Building on that work, the quantum computing group at IBM scaled the experiment to larger molecules, which made the cover of Nature last month.

Today, we announce the release of OpenFermion, the first open source platform for translating problems in chemistry and materials science into quantum circuits that can be executed on existing platforms. OpenFermion is a library for simulating the systems of interacting electrons (fermions) which give rise to the properties of matter. Prior to OpenFermion, quantum algorithm developers would need to learn a significant amount of chemistry and write a large amount of code hacking apart other codes to put together even the most basic quantum simulations. While the project began at Google, collaborators at ETH Zurich, Lawrence Berkeley National Labs, University of Michigan, Harvard University, Oxford University, Dartmouth College, Rigetti Computing and NASA all contributed to alpha releases. You can learn more details about this release in our paper, OpenFermion: The Electronic Structure Package for Quantum Computers.

One way to think of OpenFermion is as a tool for generating and compiling physics equations which describe chemical and material systems into representations which can be interpreted by a quantum computer1. The most effective quantum algorithms for these problems build upon and extend the power of classical quantum chemistry packages used and developed by research chemists across government, industry and academia. Accordingly, we are also releasing OpenFermion-Psi4 and OpenFermion-PySCF which are plugins for using OpenFermion in conjunction with the classical electronic structure packages Psi4 and PySCF.

The core OpenFermion library is designed in a quantum programming framework agnostic way to ensure compatibility with various platforms being developed by the community. This allows OpenFermion to support external packages which compile quantum assembly language specifications for diverse hardware platforms. We hope this decision will help establish OpenFermion as a community standard for putting quantum chemistry on quantum computers. To see how OpenFermion is used with diverse quantum programming frameworks, take a look at OpenFermion-ProjectQ and Forest-OpenFermion - plugins which link OpenFermion to the externally developed circuit simulation and compilation platforms known as ProjectQ and Forest.

The following workflow describes how a quantum chemist might use OpenFermion in order to simulate the energy surface of a molecule (for instance, by preparing the sort of quantum computation we described in our past blog post):
  1. The researcher initializes an OpenFermion calculation with specification of:
    • An input file specifying the coordinates of the nuclei in the molecule.
    • The basis set (e.g. cc-pVTZ) that should be used to discretize the molecule.
    • The charge and spin multiplicity (if known) of the system.
  1. The researcher uses the OpenFermion-Psi4 plugin or the OpenFermion-PySCF plugin to perform scalable classical computations which are used to optimally stage the quantum computation. For instance, one might perform a classical Hartree-Fock calculation to choose a good initial state for the quantum simulation.
  2. The researcher then specifies which electrons are most interesting to study on a quantum computer (known as an active space) and asks OpenFermion to map the equations for those electrons to a representation suitable for quantum bits, using one of the available procedures in OpenFermion, e.g. the Bravyi-Kitaev transformation.
  3. The researcher selects a quantum algorithm to solve for the properties of interest and uses a quantum compilation framework such as OpenFermion-ProjectQ to output the quantum circuit in assembly language which can be run on a quantum computer. If the researcher has access to a quantum computer, they then execute the experiment.
A few examples of what one might do with OpenFermion are demonstrated in ipython notebooks here, here and here. While quantum simulation is widely recognized as one of the most important applications of quantum computing in the near term, very few quantum computer scientists know quantum chemistry and even fewer chemists know quantum computing. Our hope is that OpenFermion will help to close the gap between these communities and bring the power of quantum computing to chemists and material scientists. If you’re interested, please checkout our GitHub repository - pull requests welcome! By Ryan Babbush and Jarrod McClean, Quantum Software Engineers, Quantum AI Team

1 If we may be allowed one sentence for the experts: the primary function of OpenFermion is to encode the electronic structure problem in second quantization defined by various basis sets and active spaces and then to transform those operators into spin Hamiltonians using various isomorphisms between qubit and fermion algebras.↩

Announcing OpenFermion: The Open Source Chemistry Package for Quantum Computers

Posted by Ryan Babbush and Jarrod McClean, Quantum Software Engineers, Quantum AI Team


“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.”
-Paul Dirac, Quantum Mechanics of Many-Electron Systems (1929)

In this passage, physicist Paul Dirac laments that while quantum mechanics accurately models all of chemistry, exactly simulating the associated equations appears intractably complicated. Not until 1982 would Richard Feynman suggest that instead of surrendering to the complexity of quantum mechanics, we might harness it as a computational resource. Hence, the original motivation for quantum computing: by operating a computer according to the laws of quantum mechanics, one could efficiently unravel exact simulations of nature. Such simulations could lead to breakthroughs in areas such as photovoltaics, batteries, new materials, pharmaceuticals and superconductivity. And while we do not yet have a quantum computer large enough to solve classically intractable problems in these areas, rapid progress is being made. Last year, Google published this paper detailing the first quantum computation of a molecule using a superconducting qubit quantum computer. Building on that work, the quantum computing group at IBM scaled the experiment to larger molecules, which made the cover of Nature last month.

Today, we announce the release of OpenFermion, the first open source platform for translating problems in chemistry and materials science into quantum circuits that can be executed on existing platforms. OpenFermion is a library for simulating the systems of interacting electrons (fermions) which give rise to the properties of matter. Prior to OpenFermion, quantum algorithm developers would need to learn a significant amount of chemistry and write a large amount of code hacking apart other codes to put together even the most basic quantum simulations. While the project began at Google, collaborators at ETH Zurich, Lawrence Berkeley National Labs, University of Michigan, Harvard University, Oxford University, Dartmouth University, Rigetti Computing and NASA all contributed to alpha releases. You can learn more details about this release in our paper, OpenFermion: The Electronic Structure Package for Quantum Computers.

One way to think of OpenFermion is as a tool for generating and compiling physics equations which describe chemical and material systems into representations which can be interpreted by a quantum computer1. The most effective quantum algorithms for these problems build upon and extend the power of classical quantum chemistry packages used and developed by research chemists across government, industry and academia. Accordingly, we are also releasing OpenFermion-Psi4 and OpenFermion-PySCF which are plugins for using OpenFermion in conjunction with the classical electronic structure packages Psi4 and PySCF.

The core OpenFermion library is designed in a quantum programming framework agnostic way to ensure compatibility with various platforms being developed by the community. This allows OpenFermion to support external packages which compile quantum assembly language specifications for diverse hardware platforms. We hope this decision will help establish OpenFermion as a community standard for putting quantum chemistry on quantum computers. To see how OpenFermion is used with diverse quantum programming frameworks, take a look at OpenFermion-ProjectQ and Forest-OpenFermion - plugins which link OpenFermion to the externally developed circuit simulation and compilation platforms known as ProjectQ and Forest.

The following workflow describes how a quantum chemist might use OpenFermion in order to simulate the energy surface of a molecule (for instance, by preparing the sort of quantum computation we described in our past blog post):
  1. The researcher initializes an OpenFermion calculation with specification of:
    • An input file specifying the coordinates of the nuclei in the molecule.
    • The basis set (e.g. cc-pVTZ) that should be used to discretize the molecule.
    • The charge and spin multiplicity (if known) of the system.
  1. The researcher uses the OpenFermion-Psi4 plugin or the OpenFermion-PySCF plugin to perform scalable classical computations which are used to optimally stage the quantum computation. For instance, one might perform a classical Hartree-Fock calculation to choose a good initial state for the quantum simulation.
  2. The researcher then specifies which electrons are most interesting to study on a quantum computer (known as an active space) and asks OpenFermion to map the equations for those electrons to a representation suitable for quantum bits, using one of the available procedures in OpenFermion, e.g. the Bravyi-Kitaev transformation.
  3. The researcher selects a quantum algorithm to solve for the properties of interest and uses a quantum compilation framework such as OpenFermion-ProjectQ to output the quantum circuit in assembly language which can be run on a quantum computer. If the researcher has access to a quantum computer, they then execute the experiment.
A few examples of what one might do with OpenFermion are demonstrated in ipython notebooks here, here and here. While quantum simulation is widely recognized as one of the most important applications of quantum computing in the near term, very few quantum computer scientists know quantum chemistry and even fewer chemists know quantum computing. Our hope is that OpenFermion will help to close the gap between these communities and bring the power of quantum computing to chemists and material scientists. If you’re interested, please checkout our GitHub repository - pull requests welcome!


1 If we may be allowed one sentence for the experts: the primary function of OpenFermion is to encode the electronic structure problem in second quantization defined by various basis sets and active spaces and then to transform those operators into spin Hamiltonians using various isomorphisms between qubit and fermion algebras.↩

Towards an exact (quantum) description of chemistry

Posted by Ryan Babbush, Quantum Software Engineer

“...nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical...” - Richard Feynman, Simulating Physics with Computers

One of the most promising applications of quantum computing is the ability to efficiently model quantum systems in nature that are considered intractable for classical computers. Now, in collaboration with the Aspuru-Guzik group at Harvard and researchers from Lawrence Berkeley National Labs, UC Santa Barbara, Tufts University and University College London, we have performed the first completely scalable quantum simulation of a molecule. Our experimental results are detailed in the paper Scalable Quantum Simulation of Molecular Energies, which recently appeared in Physical Review X.

The goal of our experiment was to use quantum hardware to efficiently solve the molecular electronic structure problem, which seeks the solution for the lowest energy configuration of electrons in the presence of a given nuclear configuration. In order to predict chemical reaction rates (which govern the mechanism of chemical reactions), one must make these calculations to extremely high precision. The ability to predict such rates could revolutionize the design of solar cells, industrial catalysts, batteries, flexible electronics, medicines, materials and more. The primary difficulty is that molecular systems form highly entangled quantum superposition states which require exponentially many classical computing resources in order to represent to sufficiently high precision. For example, exactly computing the energies of methane (CH4) takes about one second, but the same calculation takes about ten minutes for ethane (C2H6) and about ten days for propane (C3H8).

In our experiment, we focus on an approach known as the variational quantum eigensolver (VQE), which can be understood as a quantum analog of a neural network. Whereas a classical neural network is a parameterized mapping that one trains in order to model classical data, VQE is a parameterized mapping (e.g. a quantum circuit) that one trains in order to model quantum data (e.g. a molecular wavefunction). The training objective for VQE is the molecular energy function, which is always minimized by the true ground state. The quantum advantage of VQE is that quantum bits can efficiently represent the molecular wavefunction whereas exponentially many classical bits would be required.

Using VQE, we quantum computed the energy landscape of molecular hydrogen, H2. We compared the performance of VQE to another quantum algorithm for chemistry, the phase estimation algorithm (PEA). Experimentally computed energies, as a function of the H - H bond length, are shown below alongside the exact curve. We were able to obtain such high performance with VQE because the neural-network-like training loop helped to establish experimentally optimal circuit parameters for representing the wavefunction in the presence of systematic control errors. One can understand this by considering a hardware implementation of a neural network with a faulty weight, e.g. the weight is only represented half as strong as it should be. Because the weights of the neural network are established via a closed-loop training procedure which can compensate for such systematic errors, the hardware neural network is robust against such imperfections. Likewise, despite systematic errors in our implementation of the VQE circuit, we are still able to learn an accurate model for the wavefunction. This robustness inspires hope that VQE may be able to solve classically intractable problems without quantum error correction.
While the energies of molecular hydrogen can be computed classically (albeit inefficiently), as one scales up quantum hardware it becomes possible to simulate even larger chemical systems, including classically intractable ones. For instance, with only about a hundred reliable quantum bits one could model the process by which bacteria produce fertilizer at room temperature. Elucidating this mechanism is a famous open problem in chemistry because the way humans produce fertilizer is extremely inefficient and consumes 1-2% of the world's energy annually. Such calculations could also assist with breakthroughs in fundamental science, for instance, in the understanding of high temperature superconductivity.

Though many theoretical and experimental challenges lay ahead, a quantum enabled paradigm shift from qualitative / descriptive chemistry simulations to quantitative / predictive chemistry simulations could modernize the field so dramatically that the examples imaginable today are just the tip of the iceberg.

Quantum annealing with a digital twist

Posted by Rami Barends and Alireza Shabani, Quantum Electronics Engineers

One of the key benefits of quantum computing is that it has the potential to solve some of the most complex problems in nature, from physics to chemistry to biology. For example, when attempting to calculate protein folding, or when exploring reaction catalysts and “designer” molecules, one can look at computational challenges as optimization problems, and represent the different configurations of a molecule as an energy landscape in a quantum computer. By letting the system cool, or “anneal”, one finds the lowest energy state in the landscape - the most stable form of the molecule. Thanks to the peculiarities of quantum mechanics, the correct answer simply drops out at the end of the quantum computation. In fact, many tough problems can be dealt with this way, this combination of simplicity and generality makes it appealing.

But finding the lowest energy state in a system is like being put in the Alps, and being told to find the lowest elevation - it’s easy to get stuck in a “local” valley, and not know that there is an even lower point elsewhere. Therefore, we use a different approach: We start with a very simple energy landscape - a flat meadow - and initialize the system of quantum bits (qubits) to represent the known lowest energy point, or “ground state”, in that landscape. We then begin to adjust the simple landscape towards one that represents the problem we are trying to solve - from the smooth meadow to the highly uneven terrain of the Alps. Here’s the fun part: if one evolves the landscape very slowly, the ground state of the qubits also evolves, so that they stay in the ground state of the changing system. This is called “adiabatic quantum computing”, and qubits exploit quantum tunneling to ensure they always find the lowest energy "valley" in the changing system.

While this is great in theory, getting this to work in practice is challenging, as you have to set up the energy landscape using the available qubit interactions. Ideally you’d have multiple interactions going on between all of the qubits, but for a large-scale solver the requirements to accurately keep track of these interactions become enormous. Realistically, the connectivity has to be reduced, but this presents a major limitation for the computational possibilities.

In "Digitized adiabatic quantum computing with a superconducting circuit", published in Nature, we’ve overcome this obstacle by giving quantum annealing a digital twist. With a limited connectivity between qubits you can still construct any of the desired interactions: Whether the interaction is ferromagnetic (the quantum bits prefer an aligned) or antiferromagnetic (anti-aligned orientation), or even defined along an arbitrary different direction, you can make it happen using easy to combine discrete building blocks. In this case, the blocks we use are the logic gates that we've been developing with our superconducting architecture.
Superconducting quantum chip with nine qubits. Each qubit (cross-shaped structures in the center) is connected to its neighbors and individually controlled. Photo credit: Julian Kelly.
The key is controllability. Qubits, like other physical objects in nature, have a resonance frequency, and can be addressed individually with short voltage and current pulses. In our architecture we can steer this frequency, much like you would tune a radio to a broadcast. We can even tune one qubit to the frequency of another one. By moving qubit frequencies to or away from each other, interactions can be turned on or off. The exchange of quantum information resembles a relay race, where the baton can be handed down when the runners meet.

You can see the algorithm in action below. Any problem is encoded as local “directions” we want qubits to point to - like a weathervane pointing into the wind - and interactions, depicted here as links between the balls. We start by aligning all qubits into the same direction, and the interactions between the qubits turned off - this is the simplest ground state of the system. Next, we turn on interactions and change qubit directions to start evolving towards the energy landscape we wish to solve. The algorithmic steps are implemented with many control pulses, illustrating how the problem gets solved in a giant dance of quantum entanglement.
Top: Depiction of the problem, with the gold arrows in the blue balls representing the directions we’d like each qubit to align to, like a weathervane pointing to the wind. The thickness of the link between the balls indicates the strength of the interaction - red denotes a ferromagnetic link, and blue an antiferromagnetic link. Middle: Implementation with qubits (yellow crosses) with control pulses (red) and steering the frequency (vertical direction). Qubits turn blue when there is interaction. The qubits turn green when they are being measured. Bottom: Zoom in of the physical device, showing the corresponding nine qubits (cross-shaped).
To run the adiabatic quantum computation efficiently and design a set of test experiments we teamed up with the QUTIS group at the University of the Basque Country in Bilbao, Spain, led by Prof. E. Solano and Dr. L. Lamata, who are experts in synthesizing digital algorithms. It’s the largest digital algorithm to date, with up to nine qubits and using over one thousand logic gates.

The crucial advantage for the future is that this digital implementation is fully compatible with known quantum error correction techniques, and can therefore be protected from the effects of noise. Otherwise, the noise will set a hard limit, as even the slightest amount can derail the state from following the fragile path to the solution. Since each quantum bit and interaction element can add noise to the system, some of the most important problems are well beyond reach, as they have many degrees of freedom and need a high connectivity. But with error correction, this approach becomes a general-purpose algorithm which can be scaled to an arbitrarily large quantum computer.

When can Quantum Annealing win?

Posted by Hartmut Neven, Director of Engineering

During the last two years, the Google Quantum AI team has made progress in understanding the physics governing quantum annealers. We recently applied these new insights to construct proof-of-principle optimization problems and programmed these into the D-Wave 2X quantum annealer that Google operates jointly with NASA. The problems were designed to demonstrate that quantum annealing can offer runtime advantages for hard optimization problems characterized by rugged energy landscapes.

We found that for problem instances involving nearly 1000 binary variables, quantum annealing significantly outperforms its classical counterpart, simulated annealing. It is more than 108 times faster than simulated annealing running on a single core. We also compared the quantum hardware to another algorithm called Quantum Monte Carlo. This is a method designed to emulate the behavior of quantum systems, but it runs on conventional processors. While the scaling with size between these two methods is comparable, they are again separated by a large factor sometimes as high as 108.
Time to find the optimal solution with 99% probability for different problem sizes. We compare Simulated Annealing (SA), Quantum Monte Carlo (QMC) and D-Wave 2X. Shown are the 50, 75 and 85 percentiles over a set of 100 instances. We observed a speedup of many orders of magnitude for the D-Wave 2X quantum annealer for this optimization problem characterized by rugged energy landscapes. For such problems quantum tunneling is a useful computational resource to traverse tall and narrow energy barriers.
While these results are intriguing and very encouraging, there is more work ahead to turn quantum enhanced optimization into a practical technology. The design of next generation annealers must facilitate the embedding of problems of practical relevance. For instance, we would like to increase the density and control precision of the connections between the qubits as well as their coherence. Another enhancement we wish to engineer is to support the representation not only of quadratic optimization, but of higher order optimization as well. This necessitates that not only pairs of qubits can interact directly but also larger sets of qubits. Our quantum hardware group is working on these improvements which will make it easier for users to input hard optimization problems. For higher-order optimization problems, rugged energy landscapes will become typical. Problems with such landscapes stand to benefit from quantum optimization because quantum tunneling makes it easier to traverse tall and narrow energy barriers.

We should note that there are algorithms, such as techniques based on cluster finding, that can exploit the sparse qubit connectivity in the current generation of D-Wave processors and still solve our proof-of-principle problems faster than the current quantum hardware. But due to the denser connectivity of next generation annealers, we expect those methods will become ineffective. Also, in our experience we find that lean stochastic local search techniques such as simulated annealing are often the most competitive for hard problems with little structure to exploit. Therefore, we regard simulated annealing as a generic classical competition that quantum annealing needs to beat. We are optimistic that the significant runtime gains we have found will carry over to commercially relevant problems as they occur in tasks relevant to machine intelligence.

For details please refer to http://arxiv.org/abs/1512.02206.

Simulating fermionic particles with superconducting quantum hardware

Posted by Rami Barends and Julian Kelly, Quantum Electronics Engineers and John Martinis, Research Scientist

Digital quantum simulation is one of the key applications of a future, viable quantum computer. Researchers around the world hope that quantum computing will not only be able to process certain calculations faster than any classical computer, but also help simulate nature more accurately and answer longstanding questions with regard to high temperature superconductivity, complex quantum materials, and applications in quantum chemistry.

A crucial part in describing nature is simulating electrons. Without electrons, you cannot describe metals and their conductivity, or the interatomic bonds which hold molecules together. But simulating systems with many electrons makes for a very tough problem on classical computers, due to some of their peculiar quantum properties.

Electrons are fermionic particles, and as such obey the well-known Pauli exclusion principle which states that no fermions in a system can occupy the same quantum state. This is due to a property called anticommutation, an inherent quantum mechanical behavior of all fermions, that makes it very tricky to fully simulate anything that is composed of complex interactions between electrons. The upshot of this anticommutative property is that if you have identical electrons, one at position A and another at position B, and you swap them, you end up with a different quantum state. If your simulation has many electrons you need to carefully keep track of these changes, while ensuring all the interactions between electrons can be completely, yet separately tunable.

Add to that the memory errors caused by fluctuation or noise from their environment and the fact that quantum physics prevents one from directly monitoring the superconducting quantum bits (“qubits”) of a quantum computer directly to account for those errors, and you've got your hands full. However, earlier this year we reported on some exciting steps towards Quantum Error Correction - as it turns out, the hardware we built isn't only useable for error correction, but can also be used for quantum simulation.

In Digital quantum simulation of fermionic models with a superconducting circuit, published in Nature Communications, we present digital methods that enable the simulation of the complex interactions between fermionic particles, by using single-qubit and two-qubit quantum logic gates as building blocks. And with the recent advances in hardware and control we can now implement them.

We took our qubits and made them act like interacting fermions. We experimentally verified that the simulated particles anticommute, and implemented static and time-varying models. With over 300 logic gates, it is the largest digital quantum simulation to date, and the first implementation in a solid-state device.
Left: Model picture with four fermionic modes in two sites. The modes are occupied or unoccupied. For example, we can start with two fermionic particles in the right well, by occupying the blue and green mode. If the particles repel each other, there's a good chance that one of the them will hop to the left well through the process of quantum tunneling through the barrier. It will then occupy the red or purple mode. This interplay of on-site interaction and hopping lies at the core of describing processes in physics and chemistry, ranging from the conductivity of metals to the binding between atoms in molecules. Right: The false-colored cross-shaped structures are the superconducting quantum bits. The colors correspond to the modes, so if we have two fermionic particles in the blue and red modes, the rightmost two quantum bits are excited.
Coming up with an efficient sequence of logic gates that can accurately model the interactions for systems of fermions wasn’t easy. So we teamed up with Dr. Lucas Lamata, M.Sc. Laura García-Álvarez, and Prof. Enrique Solano from the QUTIS group at the University of the Basque Country (UPV/EHU) in Bilbao, Spain, who are experts in constructing algorithms and translating them into the streams of logic gates we can implement with our hardware.

For the future, digital quantum simulation holds the promise that it can be run on an error-corrected quantum computer. But before that, we foresee the construction of larger testbeds for simulation with improvements in logic gates and architecture. This experiment is a critical step on the path to creating a quantum simulator capable of modeling fermions as well as bosons (particles which can be interchanged, as opposed to fermions), opening up exciting possibilities for simulating physical and chemical processes in nature.