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Quantum errors can be corrected

Classical computers store and process information as “bits” that can have one of two values – “0” or “1”. A quantum computer, on the other hand, would exploit the ability of quantum particles to be in “superpositions” of two or more states at the same time. This means that a quantum computer could, in principle, outperform a classical computer for certain tasks.

However, quantum bits or “qubits” are very fragile and any noise in a quantum computer could change the state of a qubit quite easily, with adverse consequences for any calculations being performed by the computer. A practical quantum computer therefore needs to be able to correct these errors. Errors also occur in classical computers but they are much less frequent.

David Wineland and colleagues at the National Institute of Standards and Technology (NIST) in Colorado begin by creating a qubit that is a superposition of two hyperfine levels in the ground state of a trapped beryllium ion. They then entangle this “primary” ion with two “ancilla” ions, which are not used in the actual computation. Entanglement allows quantum particles to have strong correlations that are not possible in classical physics. In the NIST experiment it means that any errors that occur in one of the ions have an affect on the other two ions.

The NIST physicists then apply an artificial error of known size to their system, before disentangling the three ions and measuring the quantum state of the two ancilla qubits. Depending on the result of this measurement, they are able to determine what needs to be done to return the primary qubit to its initial state.

“The primary qubit can be corrected and restored as if an error never happened,” says John Chiaverini of NIST. “In principle, this method can be repeated indefinitely, which is a necessity of large-scale computation.” Another challenge is to correct all sorts of errors and not just the specific spin rotation applied by the NIST team.

Hot on the trail of Copernicus

The world has never lacked seekers of wisdom, however arduous to acquire; seekers of knowledge, however challenging to learn; and seekers of novelty, however threatening to appropriate. This seems to be the fundamental message from Owen Gingerich’s three decades-long study of the reception of and response to Nicolaus Copernicus’s epoch-making book De Revolutionibus (Nurenberg 1543 and Basel 1556). Professionally planned and meticulously executed, The Book Nobody Read is Gingerich’s attempt to share this exploration, which for him was more a personal odyssey than an academic investigation. It is a book about a book and the history of that book.

The genesis of Gingerich’s project is also equally remarkable and unusual. It began in 1959 when Arthur Koestler wrote in his highly acclaimed best-seller The Sleepwalkers that “The Book of the Revolutions of the Heavenly Spheres was and is an all-time worst-seller”. In fact, Koestler called it “the book that nobody read”. This was no casual, inadvertent remark. He substantiated his claim with data and details, which included a comparison of the circulation and reception of similar books of the era. The praise showered on The Sleepwalkers by reviewers added substantial weight to Koestler’s claim.

Although the scholarly world initially agreed with Koestler’s seemingly scholarly surmise, there lingered a certain unease in the minds of serious, reflective scholars. Could De Revolutionibus – a book that changed the course of the history of science, if not of history itself – have been stillborn? Could a book have brought about such a turnaround in our ideas without the active acceptance and appreciation of the scholars of the day? A negative answer in this context is tantamount to believing that scientific revolutions take place by sheer fluke. Thoughts of this nature must have left Gingerich and other like-minded scholars with a certain disquiet, and provoked them to look at the matter anew.

An opportunity presented itself in 1970 when Gingerich stumbled upon a first-edition copy of De Revolutionibus while exploring a huge safe full of rare astronomy books at the Royal Observatory, Edinburgh. On close examination, he found that this was no ordinary copy of a rare book; it was heavily annotated by none other than Erasmus Reinhold, the leading astronomy teacher of northern Europe in the 1540s.

Questions filled Gingerich’s mind. Do any other copies still survive? Where can they be located? What can they tell us about Copernicus’s new theory or about the reception accorded to De Revolutionibus by the leading scholars and thinkers of the day? This was the genesis of The Book Nobody Read and its forerunner An Annotated Census of Copernicus’ De Revolutionibus.

Gingerich thus began a chase for original copies of the book that took him to centres of learning around the globe – from Aarhus to Beijing, from Coimbra to Dublin, from Melbourne to Moscow, and from St Gallen to San Diego. The present book is Gingerich’s “personal memoir” about the making of the census.

The theme of the book at first sight may appear drab and dry – a humdrum secretarial task of collecting and cataloguing the first and second editions of De Revolutionibus. It may even look too ordinary to engage an accomplished scholar and expert. But far from being a dead catalogue of copies, it becomes a scholarly study of a book that is full of life.

Gingerich’s book unfolds into a masterly study of the history of astronomy and of various hard astronomical theories. It includes little known facts about well-known personalities, should-know facts about little-known persons, as well as accounts of destructive tensions and constructive collaborations between scholars and experts. In fact, one is led to conclude that a project of this diversity and magnitude could have been accomplished only by a scholar of wide knowledge, deep insights and passionate commitment.

Gingerich’s persistent, professional, indefatigable, decades-long chase brought rich dividends and Koestler was proven dead wrong. Far from being a book that nobody read, De Revolutionibus, which discussed whether the Earth orbits the Sun, was seriously studied by many outstanding astronomers of the day, as revealed by the extensive and erudite marginalia on many copies. The expert research on these marginalia becomes a study of how the new Copernican perspective grew in the minds of contemporary scholars and how it was gradually accepted by the intelligentsia of the day.

Copies of the first editions were owned by saints like the Jesuit Aloysius Gonzaga, heretics like Giordano Bruno, scholars like John Dee and Thomas Diggs, and geniuses like Tycho Brahe, Galileo and Kepler. Kings, musicians, medicine men and bibliomaniacs even boasted of treasuring it. The hitherto incomplete list in the census has already traced over 600 copies.

Autobiographical in style, the book brings to light much information and insights, some for the first time, at least for non-experts. Tucked inside the pages are pieces of information like Galileo’s “little taste for the details of celestial mechanics”, Bruno’s unfamiliarity with important Copernican ideas, the explanation for the “Tower of the Winds” at the Vatican, and an account of Tycho Brahe’s marriage to a woman of unequal class. The long story about Gingerich’s tracing of the Vatican copy of Copernicus reveals a feat of scholarly, smart sleuthing.

In recounting the story of his own peregrinations, the author takes readers on an informative and exciting tour of the most prestigious libraries of the world, and gives an insight into the politics and intrigue that accompany such operations. Weaved into the book is a great deal of historical, scientific, sociological, religious and cultural information. Another noteworthy feature is the author’s extensive, detailed and painstaking study done on the marginalia, which has yielded many valuable and otherwise inaccessible insights.

Although many careful readers may find the occasional repetitions in the book unnecessary and distracting, those who are less familiar with astronomy and the history of science may find them helpful. This book will therefore appeal to many different readers, including astronomers, historians, book-lovers and biographers. With many personal and professional anecdotes, plus witty and critical observations, the text reads like a detective novel. I would re-christen Gingerich’s book as “The Book Everybody Could Read” – with much delight and great profit.

Physics for taxi-drivers

Most people enjoy conversations with taxi-drivers. I occasionally fantasize the driver beside me morphing into my father, who was a London cabbie and never shy about sharing his opinions with his fares. A recent journey began with a question: “I often drive between the airport and the university physics department, but I don’t understand what you do there. It seems very clever, but does it have anything to do with the real world?” My response was to tell the driver two stories.

The first began with a factory producing consumer goods – surely as connected with “the real world” as anyone who uses that dismal expression would wish. This particular factory makes CD players, which enable anyone to hear music, reproduced almost perfectly, anywhere in the world: in the desert, up mountains, in forests, on the seas and so on. This is a new development in human history. Previously, people who wanted to hear music had to be physically present when and where it was performed (or, more recently, within range of radio reception and tuning in at the right time). In a sense, the CD player represents the ultimate cultural democracy: making available to many what could previously be enjoyed only by a few.

But the CD player is also a “quantum physics machine”, in several senses. First, it contains a laser, the beam of which reflects from the pits on the disk that encode the music. This was an application by engineers of a device invented 30 years earlier, and, moreover, an unanticipated application – recall the cliché that the laser was an invention looking for an application. The laser itself was an application by physicists of ideas about the quantum physics of light, initiated by Einstein four decades earlier still – an application that was surely not anticipated by Einstein himself. And, of course, each of the millions of transistors that guide the flow of electrical information is itself a direct application of the quantum mechanics of electrons in the periodic environment of a crystal.

From this chain of associations comes an unexpected conclusion: quantum mechanics has democratized music.

Note the connection between three abstractions: quantum mechanics, which raises doubts about previous conceptions of physical reality; democracy, a relatively recent ideal in the organization of human affairs; and music, where sounds unrelated to any in the natural world, and continually changing in ways incomprehensible to each previous generation, exert strange powers over our deepest emotions.

Where and when

For my second story, I pointed to the taxi’s GPS navigation device. To decode the signals from the GPS satellites to discover our location, it is necessary to incorporate various relativistic effects that influence the time it takes for the signals to travel to and from the satellites. If these effects were ignored, disastrous navigation errors would soon result. So the GPS is a “relativity physics machine” – to my knowledge, the first such consumer device. It enables the solution, for all everyday purposes, of the ancient problem of knowing where we are, much as the vibrations of little quartz crystals in wristwatches recently solved the analogous problem of knowing when we are.

I have deceived you. The driver’s question caught me off balance, and what he got from me was not what you have just read but ill-articulated and probably not very convincing versions of these two stories. But they did seem to give him an unfamiliar perspective on what we do, because he went on to ask me about my own research in theoretical physics, which has no applications as far as I know. (Actually it does have applications in philosophy, where it contributes to the long-standing problem of how our different levels of descriptions of the physical world – e.g. classical and quantum mechanics – are related.)

However, physicists should avoid the serious misjudgement of claiming these revolutionary devices for ourselves, as though we alone created them. No physicist, or group of physicists, would ever have produced a CD player. As I have already mentioned, the invention of the CD player also involved engineers, while mathematicians developed and optimized the codes that transform the music into light and electricity. Moreover, to get any invention into people’s hands requires factories to produce it, businesses to finance and sell it, advertisers to tell people about it and so forth. The world is strangely and wonderfully connected.

Nevertheless, it is good for people to see that our subject affects them in ways they usually have no conception of. Not just taxi-drivers, of course: I made the CD player story the centrepiece of a recent prize-giving speech at a secondary school, and it seemed to go down well with the 11-18-year-old students in the audience.

Be prepared

Now I come to the point. 2005 is the International Year of Physics, when we can hope for a higher profile for our subject. We should anticipate more questions like that of my taxi-driver, and be ready with a good supply of convincing stories. As it happens, both of mine involve Einstein, which is appropriate for the year that also celebrates the centenary of his monumental early contributions. Every physicist knows dozens more.

My reason for writing this article is to encourage readers to share their favourite stories by sending them to this magazine, which will publish a selection of the best. The type of stories I have in mind are characterized by connections between the intellectual and the everyday, where an abstract concept in physics leads, perhaps after many decades, to a direct effect on people’s lives. Stories should be not more than 500 words and should be sent to Physics World before 31 January 2005.

• This competition has now closed. The winning entries will be published in the March 2005 issue of Physics World

Shelf life: Lee Smolin

What are the three best popular science books?

Relativity: The Special and General Theory by Albert Einstein because it is readable by a child who knows no maths. If you read the book carefully and think about each page as you do so you really undergo an intellectual transformation and come out understanding relativity. I have used it in several courses for non-scientists and it works.

The Selfish Gene by Richard Dawkins introduced me to the real logic of natural selection, and makes an extremely strong case for it.

Microcosmos by Lynn Margulis floored me. It radically changed my way of thinking about biology and life, and led to new ideas in my own work.

All of these books make an important scientific and philosophical argument in language that laypeople can understand. However, they are also the real stuff – being written by scientists who are constructing a case for something they have understood and passionately believe in. These books show that science is indeed focused and disciplined common sense, and that any important scientific idea can be explained and argued for in careful, non-technical language.

What books are you currently reading?

I am reading Bright Galaxies, Dark Matter – a collection of essays by Vera Rubin, the American astronomer who discovered that the motions of stars in galaxies cannot be accounted for by Newton’s laws unless there is a lot of hidden, or dark, matter. Rubin is one of the great living scientists and there is no more influential or consequential discovery in the astronomy and physics of the last 40 years. Indeed, we still do not know what dark matter is – or even if it is dark matter rather than a breakdown in Newton’s law of gravity.

The fact that Rubin made this discovery as a young scientist – working alone or with a few students – teaches us that the individual judgement and initiative of a scientist, choosing her own direction, is still the most reliable way for science to progress. The story she tells also takes us back to the days when women were excluded from many areas of academic life. In 1965 she was the first woman permitted to observe at the Palomar observatory. We have come a long way since then, but the fact that Rubin is not a household name perhaps tells us that we have still a long way to go.

Which popular-science book have you never read, but feel you ought to have tackled?

I must confess I never made it more than a few pages into Stephen Hawking’s A Brief History of Time because I found the reasoning about things like quantum cosmology, imaginary time and so on as confusing and as sloppily argued in the book as in the original papers. If I can’t understand what it could mean to say that time “becomes imaginary” at “early times” – and I am a professional who works in the same field – how is the public supposed to understand what is meant?

I tried to read the book in the hope that Hawking used the opportunity to explain the idea without the technicalities, in a way that I could perhaps finally understand, but ended up feeling that there is possibly really nothing there. However, I have always felt guilty about not finishing the book, and wonder that if I persevered through the bad prose I might finally understand what he is proposing.

The book not read

In his book about a book, Gingerich tells the story of his attempts to track down early copies of De Revolutionibus, the book in which Nicolaus Copernicus proposed that it is the Sun, not the Earth, that is the centre of the solar system. In particular Gingerich investigates claims that no-one actually read De Revolutionibus and finds these claims to be without foundation. Indeed, it seems as if anyone who was anyone in astronomy at the time – Tycho Brahe, Galileo and Kepler – read Copernicus’s classic.

Although few of us would feel the need to read De Revolutionibus today, there is a vast and ever-increasing number of popular or semi-popular books aimed at both scientists and non-scientists – books that the enlightened reader might feel they should at least buy or borrow and, if time permits, actually read. This writer’s shelves are full of such volumes.

When selecting books for review in Physics World we try to avoid titles that will only be of interest to a minority of readers, which means that we tend towards popular, biographical and historical books, and away from monographs and specialist titles. Even then there are more suitable titles than we have pages to review, so we have introduced a new column called “Between the lines” to bring these books to the attention of readers. Also new is “Shelf life”, a column where leading physicists tell us what they have and have not been reading.

Publishers are notoriously secretive about the sales figures for their books so it is difficult to know whether the growing numbers of popular-science books are indeed popular, although a group of statistical physicists has recently reached some interesting conclusions about the sales of books by analysing data from Amazon.com (Sornette et al. 2004 Phys. Rev. Lett. at press). One wonders if there really can be a demand for yet another popular book with an astronomical theme or another collection of essays about breakthroughs and disputes in modern science. However, there are certainly plenty of these books on sale in high-street bookshops and one must assume that publishers know what the public is likely to buy.

Often the challenge for the physicist-reader is finding a popular book that contains enough new material. By necessity popular books about the very latest breakthroughs must include enough background material for the layreader, but one can only read about the basics of quantum mechanics or special relativity so many times. The best of these books also manage to go behind the scenes and reveal the personalities and motivations of those involved, and ensure that they end up being read rather than gathering dust on the shelf.

Turbulent transition for fluids

Turbulent flows, in which volumes of fluid seem to zigzag independently among random eddies, are ubiquitous in nature. They occur in the interstellar gas and in the airways to our lungs. Turbulence can also be expensive. If the flow of water or gas in a pipe is turbulent, rather than laminar, more pressure is needed to maintain the same volume of discharge. And the faster a car or an airplane moves, the more turbulence will hinder the motion and increase fuel costs.

As physicists, we would like to have a theory that explains this phenomenon. But a complete understanding of turbulence in ordinary classical fluids remains elusive. Now Frans Nieuwstadt of the Delft University of Technology in the Netherlands and co-workers have performed an experiment that could take us closer to this goal. By studying the flow of water in a long pipe, the team has observed travelling-wave patterns during the transition from smooth to turbulent flow for the first time. Moreover, the results are in good agreement with recent predictions (B Hof et al. 2004 Science 305 1594).

Reynolds’ puzzle

The problem of turbulence goes back to 1883, when Osborne Reynolds of Manchester University made a remarkable discovery that has remained a puzzle ever since. By injecting a small amount of ink into a horizontal glass pipe filled with water, he was able to check whether the flow was laminar or turbulent. Reynolds found that the transition from laminar to turbulent flow occurs spontaneously if a dimensionless quantity, R, is larger than 2000.

This quantity, which is known as the Reynolds number, is defined as the ratio of the inertial and viscous forces on the fluid: R = VD/v, where V is the mean velocity of the flow, D is the diameter of the pipe and v is the viscosity of the fluid. We now know that Reynolds’ initial figure of 2000 varies from experiment to experiment. Indeed, by carefully controlling the flow at the inlet of the pipe, laminar flows can be sustained at Reynolds numbers up to 100,000.

From direct observations of fluids like those carried out by Reynolds, researchers know that the profile of a fluid during laminar flow is parabolic. This is also clearly seen by solving the Navier-Stokes equation, which forms the basis of classical fluid dynamics. So why does the flow suddenly become unstable and break up into turbulent swirls at large enough values of R?

The standard way to answer this kind of question is to study the response of a fluid when it is subject to infinitesimally small disturbances, using a mathematical tool called linear stability theory. For example, imagine carefully placing a ball at the top of a perfectly smooth hill. If we gently nudge the ball, it will roll down away from the top. On the other hand, if we place the ball in a valley between hills and then perturb it, it will come back to its rest position. We therefore conclude that the bottom position is stable, while the top position is unstable.

Infinitesimal instabilities

When applied to the infinite degrees of freedom in a fluid, linear stability theory predicts a great variety of phenomena – from thermal convection to vortex formation in rotating flows. Typically, one begins with a simple “primary” solution of the Navier-Stokes equation and then tracks this solution while increasing some parameter that drives the disturbances. At some critical value of the drive parameter, the primary solution becomes unstable and a new solution branches out with a more complex flow pattern, a process called bifurcation.

In the case of a fluid in a pipe, the primary solution is the parabolic profile and the drive parameter is R. Unfortunately, the theory fails miserably in predicting any bifurcation: the parabolic profile is always stable, even if R becomes infinite! This direct contradiction of Reynolds’ experiment is surprising, since the tiny disturbances in linear stability theory represent the mechanical vibrations and thermal noise that are unavoidable in any physical system. Indeed, even computer simulations are subject to such disturbances as a result of rounding and truncation errors. So what is so special about a fluid in a pipe?

Since infinitesimal perturbations do not appear to trigger turbulence, the culprit must be perturbations with a larger amplitude. Indeed, Tom Mullin of Manchester University and co-workers, including Björn Hof, who was also involved in the Delft experiment, recently repeated Reynolds’ experiment with extraordinary care, and found that the amplitude of the perturbations required to establish turbulence scales inversely with R (B Hof et al. 2003 Phys. Rev. Lett. 91 244502).

Mathematically, such finite amplitudes are bad news because they mean that one has to face the full nonlinearity of the Navier-Stokes equation. But some applied mathematicians have risen to this challenge. Richard Kerswell of Bristol University in the UK, Fabian Waleffe of the University of Wisconsin and Bruno Eckhardt of Philipps University in Marburg, Germany – who also participated in the Delft experiment – have discovered a new class of travelling-wave solutions to the Navier-Stokes equation that exist for values of R above approximately 1200. These solutions are unstable since they are not related to the primary branch of the parabolic profile, and obtaining them required a mathematical tour de force.

Kerswell and co-workers mapped the pipe scenario onto a more general problem with an additional drive parameter, for which a bifurcation can be found. The new solution branch was then tracked back to the point where the second drive parameter vanishes, leaving a new solution to the original pipe-flow problem that is disconnected from the primary parabolic profile. The new, unstable travelling-wave solutions consist of flow-wise swirls and streaks with rotational symmetry about the axis of the pipe. It now seems that these solutions have been observed in an experiment.

Experimental solution

To investigate the transition from laminar to turbulent flow, Nieuwstadt and co-workers injected water into a pipe 26 m in length. They then studied the region of turbulent flow that was produced further down the pipe using a particle image velocimeter (a combination of high-speed cameras and pulsed lasers) to measure the whole velocity field across the flow at very high repetition rates. Impressively, the Delft team observed transient flow structures with 2-, 3-, 4- and 6-fold symmetry that are very similar to the new travelling-wave solutions (see figure).

This transition to turbulence can be likened to pinball. The flow undergoes erratic behaviour whereby it moves from one unstable solution to another before returning to the primary parabolic profile. This happens because more travelling waves appear as R is increased, which can attract trajectories from most other directions in phase space. The unstable waves form an attracting region of trajectories that becomes larger and larger, whereas the basin of attraction of the parabolic profile shrinks. In other words, perturbations of smaller amplitude become large enough to trigger turbulence in agreement with Mullin’s finding.

This breakthrough is the result of a remarkable collaboration between applied mathematicians and physicists. The feeling of wonder inspired by watching a turbulent flow can only be matched by the breadth of application and predictive power of the Navier-Stokes equation. Although this short, innocent-looking equation was devised 150 years ago, it seems we still have to unlock all of its secrets.

Superconducting quantum bits

The quantum world looks very different to the ordinary world. A quantum particle can, for instance, be in two places simultaneously, while its speed and position cannot both be measured with complete accuracy at the same time. Moreover, if its mass is small enough, a quantum particle can tunnel through energy barriers that its classical counterparts could never cross.

Physicists are comfortable with the use of quantum mechanics to describe atomic and subatomic particles. However, in recent years we have discovered that micron-sized objects that have been produced using standard semiconductor-fabrication techniques – objects that are small on everyday scales but large compared with atoms – can also behave as quantum particles.

These artificial quantum objects might one day be used as “quantum bits” in a quantum computer that could perform certain computational tasks much faster than any classical computing device. Before then, however, these devices will allow us to explore the interface between the quantum and classical worlds, and to study how interactions with external degrees of freedom lead to a gradual disappearance of quantum behaviour.

Quantum information

In classical computers information is stored and processed as “bits” that can have one of two possible values: “0” or “1”. Quantum bits or “qubits” are different: in addition to “0” and “1” a qubit can also exist in a superposition state, α|0〉 + β|1〉, in which it can have both values at the same time. However, when a qubit is measured, the result is either “0”, with a probability of α², or “1”, with a probability of β².

Such probabilistic behaviour hardly seems to be a good basis for processing information. However, for as long as we can avoid making measurements the quantum system will evolve in a completely deterministic way and, moreover, will maintain its ability to be in two places at the same time and so forth. This ability to remain quantum rather than classical over a period of time is called quantum coherence. Here the word “measurement” refers to events, both intentional and accidental, in which quantum information is transferred out of the quantum system. In the absence of any such measurements the system maintains its quantum coherence.

It is also possible to create a state that is even stranger than a normal quantum state by “entangling” two or more quantum particles. Such an entangled state was demonstrated for the first time with photons in the early 1980s. If the quantum state of one of the entangled particles is changed (e.g. if the polarization of a photon is changed), then the state of the other particle instantly changes in a certain way, even if there is no interaction between them. Quantum computing is based on these three properties of quantum mechanics: superposition, coherence and entanglement (see Physics World March 1998 pp33-57).

To be useful a quantum computer will need at least 10,000 qubits. However, it is very difficult to find a technology that combines the necessary level of control over two-level quantum systems with the possibility of mass fabrication. Researchers can routinely create large ensembles of nuclear spins using magnetic resonance techniques, but it is almost impossible to control individual spins.

Figure 1

Single ions and atoms in cavities and traps can be controlled with exquisite precision, but it will be difficult to scale up these techniques to thousands of atoms or ions. Solid-state qubits, on the other hand, have only just been demonstrated, but the microfabrication tools used in the semiconductor industry could provide a route to mass fabrication.

Superconducting qubits

To create a solid-state qubit, like any other kind of qubit, we need to isolate a two-level quantum system. To date, efforts to make solid-state qubits have focused on superconductors and semiconductors. While interesting results have been obtained with two semiconductor approaches – quantum dots and single-donor systems – the superconducting approach is currently the most advanced.

To maintain coherence it is essential to keep electron-electron interactions, and also interactions between electrons and other degrees of freedom (such as phonons in the solid), under control. Superconductors have the advantage in this regard because the electrons condense into Cooper pairs that form a single superfluid. This superfluid is able to move through the metal lattice without any resistance (i.e. without interactions) because it takes a certain amount of energy, known as the energy gap, to break up the Cooper pairs.

In aluminium – a popular material for making superconducting quantum circuits – the energy gap corresponds to a frequency of 90 GHz at a temperature of 20 mK. This gap is an order of magnitude greater than the typical energy difference between the two levels in a superconducting qubit, which means that we can “drive” the qubit without breaking up the Cooper pairs and jeopardizing the quantum coherence of the system.

The behaviour of the electron superfluid is completely determined by a single quantum wavefunction. The amplitude of this wavefunction determines the number of Cooper pairs, while the value of the phase is related to the supercurrent and any magnetic field that is present. The amplitude and phase of the wavefunction are conjugate variables – that is, they are related by an uncertainty principle that means we cannot measure both of their values with arbitrary precision at the same time. The two primary types of superconducting qubit, the charge qubit and the flux qubit, are directly related to these two variables: charge qubits are associated with the amplitude, while flux qubits are related to the phase.

A key component in most superconducting qubits is a device called a Josephson junction. This consists of a thin layer of aluminium oxide, which is an insulator, sandwiched between two superconducting layers of aluminium, which becomes a superconductor when cooled below 1.2 K. The insulating layer is so thin (a few nanometres) that Cooper pairs can tunnel through it and couple the superconducting wavefunctions on either side of the barrier. Most of the circuits for superconducting qubits built so far consist of Josephson junctions and other components like capacitors connected by superconducting leads made of aluminium.

Different types of qubits

Two energies are important when designing a superconducting qubit. The Josephson energy, E_J, is a measure of the strength of the coupling across the junction, while the Coulomb charging energy, E_C, is the energy needed to increase the charge on the junction by 2e. The junction is essentially a capacitor with E_C = 4e²/2C, where e is the charge of the electron and C is the capacitance.

Figure 2

If E_C is larger than E_J, the circuit tends to fix the numbers of Cooper pairs. However, tunnelling through the junction can lead to transitions between states containing different numbers of Cooper pairs. This combination creates the charge qubit.

On the other hand, if E_J is larger than E_C, the requirement that the total phase difference around a closed loop in the circuit must be a multiple of 2π dominates, and the circuit can be used as a flux qubit. With the proper choice of parameters, transitions between phase states – e.g. from 0 to 2π – are possible.

The charge qubit consists of a small volume of superconductor, known as a Cooper-pair box, that is connected to a weak Josephson junction and driven with a gate voltage, V_g, through a capacitor. If V_g = e/C_g, where C_g is the capacitance of the gate, then the classical states in which there are zero (|0>) and one (|1>) extra Cooper pairs in the box have the same energy (see figure 1a).

However, quantum tunnelling through the Josephson junction results in the formation of two new quantum states: one is a symmetric superposition of the classical zero and one states (|0> + |1>), while the other is an antisymmetric superposition (|0> – |1>). These new quantum states differ in energy by 2E_J, and this superposition forms the basis of the charge qubit (see figure 1b). By applying time-varying signals to the voltage gate it is possible to control the dynamics of the system.

A typical flux qubit is made by joining three Josephson junctions with superconducting leads to form a closed loop and using an applied magnetic field (perpendicular to the loop) to drive the circuit by controlling the phase (see figure 1c). The junctions are needed to provide a “weak spot” where transitions between different phase states can take place. A flux qubit could be made with just one or two junctions, but the use of three allows the behaviour of the circuit to be fine-tuned more easily.

When the magnetic flux through the loop, Φ, is equal to half the quantum of magnetic flux in a superconductor, the state with a zero phase difference around the loop (|0〉) has the same energy as the state with a 2π phase difference (|1〉). One of these states corresponds to a current circulating around the loop in the clockwise direction, while the other state corresponds to a current moving in the opposite direction. As happens with the charge qubit, two new quantum superposition states are formed, and again the difference in energy between the states is equal to the tunnelling strength, which depends on a number of parameters (see figure 1d).

There is also a third type of superconducting qubit, usually known as a phase qubit. This employs a single Josephson junction and the two levels are defined by quantum oscillations of the phase difference between the electrodes of the junction.

One problem with superconducting qubits is that there are always more than two levels in the system. There can, for instance, be two or three extra Cooper pairs in the box in a charge qubit, or there can be a 4π or 6π phase difference in a flux qubit. However, with careful design the energy difference between the two “good” levels and the other “bad” levels can be made large enough to prevent quantum information leaking out of the qubit.

But there are more serious limitations. An isolated qubit may be very coherent, but this coherence can be destroyed by the connections with the outside (classical) world that are needed to control and measure the qubit. In our experiments we design the connecting circuit to optimize coherence. In principle, driving a circuit is easier than measuring it because signal sources tend to be strong, while the qubit signal is usually very weak. Strong coupling makes measurement easier, but it is also detrimental to quantum coherence. Therefore the experimentalist, guided by theory, has to make their own choices.

How to make a charge qubit

The first superconducting qubit to demonstrate its potential was the charge qubit. In a groundbreaking experiment in 1999, Yasunobu Nakamura and co-workers at NEC in Japan made a charge qubit that consisted of a small Josephson junction and an aluminium box for the Cooper pairs.

Figure 3

The NEC team started with the gate voltage below e/C_g, which meant that the box was in the “zero” state. The box actually contained millions of Cooper pairs in the zero state, which is electrically neutral, while the “one” state contains an extra Cooper pair. Next the researchers suddenly increased the voltage to e/C_g to create the state that is a superposition of these two basic charge states. Since the initial state of the system, the zero state, was not one of the new eigenstates, the system oscillated between the two superposition states with a frequency of 2E_J/h, where h is Planck’s constant (see Nakamura et al. in Further reading).

After a certain time the gate voltage was reduced again, and the system was projected back into one of the two basic charge states. By adding a small additional junction to the box it was possible to determine if the system ended up in the zero or one state, because a single non-superconducting electron escaped though this junction every time the system ended up in the one state. The readout signal was a current that depended on the relative weighting of the zero and one states. By performing the experiment many times, Nakamura and co-workers were able to show that the probability of the system returning to the zero or one state oscillated as the length of the voltage pulse increased.

The NEC group recently expanded this type of experiment to two coupled charge qubits, and was able to demonstrate the basic two-qubit quantum operations (see Yamamoto et al. in Further reading). Unfortunately, these types of qubits and the technique used to measure them cannot be easily scaled up, and the readout process limits the length of time over which quantum operations can be performed. Moreover, the purity of the charge qubit is severely limited by random fluctuations of charged defects in or near the barriers of the junctions.

However, it is possible to scale up the qubit designs based on charge states, as Daniel Estève, Michel Devoret and colleagues at the CEA laboratory in Saclay near Paris demonstrated in 2002. They developed an ingenious type of qubit, which they called the quantronium, with a readout that is completely different to that used by the NEC team.

In the quantronium the Josephson junction of the Cooper-pair box is split into two small parallel junctions (see figure 2 and Vion et al. in Further reading). Calculations show that when there is no extra charge in the box, there is a small clockwise current in the loop formed by the two junctions; and when there is one extra charge in the box, the current is anticlockwise. This means that the current can be measured, rather than the charge, and the device can therefore be described as a charge qubit with a phase readout.

To make measurements the Saclay team arranged its circuit so that the circulating current passed through a third, larger Josephson junction. The two qubit states can be distinguished by measuring the maximum current through this junction when there is no applied voltage. The Saclay team then irradiated the quantronium with microwave photons at a frequency that matched the energy difference between the two qubit states. Estève, Devoret (who is now at Yale) and co-workers confirmed that the microwaves drove what are known as Rabi oscillations between the two states at a frequency that was proportional to the amplitude of the microwave signal.

They also used magnetic-resonance techniques to determine the characteristic times for decoherence and to find “magic points” – values of the gate voltage and the flux in the current loop for which the coherence was significantly stronger. The Saclay team is now studying two-qubit systems.

Working with flux qubits

Our group at Delft in the Netherlands, in collaboration with Terry Orlando and colleagues at the Massachusetts Institute of Technology in the US, has focused on the flux qubit because the magnetic noise in practical circuits is much smaller than the electrical noise generated by defects. We use the standard flux-qubit circuit – three Josephson junctions connected by superconducting aluminium leads – with a magnetic flux, Φ, that is equal to about half the quantum of magnetic flux in a superconductor, Φ₀/2 = h/4e. The two basic states are a clockwise current in the loop (the zero state) and an anticlockwise current (the one state).

Measurements are made with a superconducting quantum interference device (SQUID). This device, which consists of two Josephson junctions in parallel, is the most sensitive magnetic-flux detector known. Moreover, SQUIDs can be fabricated with the same tools that are used to make the qubits themselves (see figures 3 and 4).

The currents in the qubits are carried by a billion Cooper pairs, and quantum tunnelling involves reversing the directions of all these particles simultaneously. Five years ago many physicists doubted if quantum tunnelling could take place in a piece of metal with dimensions of several microns. In 2000, however, experiments by Jonathan Friedman and co-workers at Stony Brook in the US and by Casper van der Wal and colleagues in Delft showed that it is possible to create a quantum superposition of distinct macroscopic states in which all the Cooper pairs are travelling in both the clockwise and anticlockwise directions at the same time (see Physics World August 2000 pp23-24).

In these experiments the absorption of microwave radiation by the circuit was measured as a function of frequency and applied flux. From these spectroscopic data it was clear that the qubit did not have two states of equal energy, but opposite current, at Φ ∼ h/4e, as would be the case if there were no quantum superposition or tunnelling. Rather, the data showed that two new symmetric and antisymmetric superposition states had been formed.

Figure 4

We went on to demonstrate Rabi oscillations and recently we performed the first spectroscopic measurements on two coupled flux qubits. Again we were able to measure energy differences as a function of the applied magnetic flux. Meanwhile John Martinis and co-workers at the National Institute of Standards and Technology (NIST) in Boulder, Colorado, have performed comparable experiments with a single-junction phase qubit. The NIST team found that decoherence in this system is mostly caused by defects in the junction barrier.

The longest decoherence times – which measure how long the system remains coherent – observed to date are between 500 ns and 4 μs, although the most important factor is the number of quantum operations that can be performed before the coherence disappears. Since a pulsed operation can be as short as about 1 ns, it should be possible, in principle, to perform hundreds or thousands of operations, although a practical quantum computer would need to be capable of performing 100,000 controlled operations.

If a superconducting qubit is well designed, then decoherence is mostly caused by materials problems such as defects, rather than the influence of the external (classical) circuit. Efforts have now began at Delft and the University of California at Santa Barbara, to where Martinis has recently moved, to use epitaxial techniques to make Josephson junctions with fewer defects.

Earlier this year Robert Schoelkopf and colleagues at Yale University in the US and, independently, Irinel Chiorescu and co-workers at Delft demonstrated coherent coupling between a qubit and a harmonic oscillator, and showed that a single photon could be transferred from the qubit to the oscillator. The ability to combine photon-based quantum communication over long distances with quantum computers built with superconducting qubits is an exciting prospect for the future of quantum information.

What next?

Superconducting qubits are not yet ideal building blocks for the quantum computers of the future. The quantum oscillations observed in experiments so far have amplitudes that are, at best, about half of what they should be. We cannot tell whether this is due to problems with the qubits or to inadequate measurements, and we have to work hard on improving both aspects.

We also need to increase decoherence times, in particular by using improved fabrication techniques to reduce the number of defects in tunnel barriers. This is not going to be easy or happen quickly. Improved fabrication techniques are also needed to improve the yield and quality of samples, which will speed up research. There is also a need for better ways to couple qubits with each other, and also with the outside world, and for new ways to switch this coupling on and off. We can expect progress in this area in the next year or two.

Progress in fabrication technology and coupling techniques will also benefit basic research. Physicists will, for instance, continue to use superconducting circuits to test fundamental aspects of quantum theory, such as entanglement and the boundary between the quantum and classical worlds.

It remains to be seen if superconducting qubits can be controlled as accurately as those based on trapped atoms or magnetic resonance, or if quantum computers built from superconducting qubits can be scaled up successfully. In theory these things are possible but it will take at least 20 years to build a real quantum computer.

It is unlikely that the superconducting quantum information processor of 2024 will have the layout that we would design now, or use the algorithms for quantum error correction or factorization that have been proposed to date. Indeed, solid-state quantum bits are as different from transistors as lasers are from light bulbs. We still have to learn how to use them and what to use them for.

Physicists tackle linguistics

All languages change over time, with some languages disappearing because they are not spoken by enough people. Stauffer and Schulze describe a particular language by a string of 8 or 16 bits, where each bit can equal 0 or 1, and start their simulations with one person speaking language zero (all bits equal to zero). Two languages are different from each other if they differ by at least one bit.

The model works as follows: After a given time, this person produces one offspring who speaks a language that might differ from that spoken by their parent by one bit: the possibility of such a mutation occurring is governed by a probability p. The model also allows for the possibility of a person dying during any iteration: this is governed by a factor called the “carrying capacity” in biology. Lastly, it is also possible that the parent decides to start speaking a different language: this is determined by several factors including the carrying capacity and the fraction of the population who already speak that language.

The Cologne physicists found that, for a sample of 10 million people, high mutation rates are needed to ensure that no single language dominates. This finding agrees with data on real languages, as does the prediction that the size distribution of languages is close to a “log-normal” distribution (see figure).

“Our model is more realistic than other similar models we know of since it allows for numerous languages, instead of only two,” say Stauffer and Schulze. “In these models, only one language survived because it was assumed to be superior to the other. We, on the other hand, have regarded all languages as being equally fit.”

However, it remains to be seen how the work will be received in the linguistics community. “Linguistics is a relatively new topic for physics and complex systems theory and any tentative way to understand and quantify it is useful and welcome,” says Marco Patriarca of the Helsinki University of Technology. “However, while the model Stauffer and Christian Schulze is interesting and worth investigating, it also seems preliminary.”

Astrophysicist thinks big

Low-energy neutrinos from the Sun have already been observed in a number of Earth-based experiments. However, high-energy neutrinos from cosmic sources are much rarer, so enormous detectors are needed to see them.

Neutrinos only interact very weakly with matter, which is both an advantage and a disadvantage in astrophysics. It means that neutrinos can travel enormous distances across the universe and through matter without losing the information they carry about their sources. However, the extreme weakness of their interactions also makes them very difficult to detect.

When neutrinos travel through large blocks of ice, they produce flashes of Cerenkov radiation when they collide with protons and neutrons in the ice. These flashes can then be detected and analysed to reveal information about the neutrino and its source. One such detector, AMANDA, is already operating in the Antarctic and can capture neutrinos with energies of up to 10¹⁵ eV. Future detectors – including IceCube and ANITA – may be able to capture neutrinos up to 10¹⁸ eV.

Gorham suggests that the large volumes of ice that exist in various parts of the solar system would be capable of detecting neutrinos at even higher energies, possibly up to 10²¹ eV. The neutrino-induced events in the ice would be monitored by an orbiting spacecraft.

“The best candidate so far is Europa, one of Jupiter’s moons, because it may have a thick covering of ice that is much larger than that found on Earth,” Gorham told PhysicsWeb. “More importantly, Europa’s ice is at about 90 Kelvin, so its thermal noise is much lower than that of Antarctic ice, which is at about 240 Kelvin.”

Axion experiment makes its debut

Although CAST did not see any axions, it improved the existing limit on the coupling between axions and photons, which is related to the mass of the hypothetical particles, by a factor of five for masses below 0.02 electron volts (arXiv.org/abs/hep-ex/0411033).

Axions are one of the leading candidates for dark matter, along with supersymmetric particles called neutralinos. They were first proposed explain why the strong interaction, unlike the weak force, does not violate charge-parity (CP) symmetry.

Axions are thought to be produced in the Sun when thermal photons scatter from electrons and protons in the solar core. Theoretical models and astrophysical observations constrain the mass of the axion to values between a millionth of an electron volt (eV) and a few eV. The detection of axions would represent further evidence for new physics beyond the Standard Model of particle physics.

If axions exist the strong magnetic field in CAST should convert axions produced in the Sun into X-ray photons. To produce such a field the CAST team recycled a test magnet that had been built for the Large Hadron Collider (LHC) at CERN. This magnet, which is 10 metres long, is capable of producing a field of 9 Tesla.

The telescope observes the Sun for around three hours every day – through one end at sunrise and the other at sunset – and points away from the Sun for the rest of the day to measure the background (figure 1).

An axion signal should appear as an excess of X-ray photons above the background in three different X-ray detectors when the Sun and magnet are aligned. Parts of the X-ray detection system were originally built for the ABRIXAS space-based X-ray telescope and have now been recycled.

Figure 2

So far the CAST collaboration has analysed data taken between May and November 2003, and is currently analysing data taken under improved conditions this year. Next year the team plans to make the experiment more sensitive at higher masses by filling the detector with a buffer gas (figure 2).

The CAST collaboration includes physicists from CERN and 14 other laboratories in Canada, Croatia, France, Germany, Greece, Russia, Spain and the US.

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