On the Road to a Quantum Computer

Ben Deen
By Ben Deen February 26, 2009 15:18

A computer in which the carriers of information behaved according to quantum mechanics could yield tremendous gains in processing speed. While today’s computers must feed through input strings one by one, quantum mechanics could allow algorithms to be executed on all possible input strings at once.

Progress toward constructing a quantum computer, however, remains in its infancy. Computations have been performed on systems of eight “quantum binary digits” (“qubits”), but these have been nowhere near modern computers’ performance.

A collaboration led by Yale professors Rob Schoelkopf, Michel Devoret, and Steven Girvin is developing a method of communication between qubits, which could be extended to systems involving more qubits working at greater distances.

Known as a “quantum bus,” this system sends information back and forth from one stationary circuit-based qubit to another via the exchange of a single photon resonating in a cavity, like a light wave bouncing between mirrors.

Alternatively, information is sent from a stationary qubit to a single traveling photon with information encoded in the probabilities of the existence and nonexistence of this moving packet of energy.

Quantum Aims and Obstacles

In quantum mechanics, observable quantities in physical systems don’t necessarily take determinate values, but are instead distributed probabilistically over a finite or infinite number of values.

The position of a particle, for instance, is given by a probability distribution over three dimensional space, not a fixed value. This critical distinction between the quantum and classical worldviews is ultimately what gives quantum computation an edge over current methods of computation.

Classical computers store information as bits, registers that hold one of two determinate values, but quantum computers store information in the probabilities that a bit holds the value 0 or 1 (known as “quantum information”).

These qubits are far harder to manage physically than classical bits: there are countless possible physical manifestations of qubits. In Schoelkopf ’s words, “Anything that is quantum mechanical—and we believe that everything is—can be a quantum computer.”

Any quantum system whose energy can take one of two discrete values is a potential qubit: for instance, the existence or nonexistence of a photon, or the alignment or antialignment of an the magnetic moment of a nucleus with an external magnetic field.

Yale researchers use circuit-based qubits: imagine a sea of electron pairs surrounding a thin barrier (about 10 atoms thick). In a superconducting circuit, a single electron pair may tunnel through the barrier to give a charge distribution of higher energy.

Why use probabilistic information at all? Superficially, this seems to just complicate matters or jeopardize the precision that we expect of a computer. While the former may be true, computing with qubits opens a range of possibilities unattainable with classical bits.

In classical computation, to see how an algorithm acts on a set of input strings, we would have to feed each one through sequentially; in quantum computation, we can take a probabilistic combination of all possible inputs and immediately obtain a combination of all corresponding outputs.

How do we extract this rich probabilistic information? When we measure a qubit’s state, we find either 0 or 1 with certain probabilities, but the qubit subsequently “collapses” to yield a single classical bit of information from each measurement.

One difficulty in quantum computation is that we must measure a large number of copies of each qubit to know its full probabilistic original state; in practice, one obtains a solution within a probability of error, which can be reduced by repeating the computation.

Luckily, clever algorithms can circumvent these issues. For instance, Shor’s factorization algorithm, the most promising quantum algorithm, factors a number into primes at a much higher speed than any algorithm running on classical bits.

Using classical bits, the time required to factor a number increases exponentially in its size. With qubits, this time can be made proportional to the log cubed of size, which increases more slowly as size increases.

However, the possibility of cracking modern cryptography methods using quantum computing remains distant: the most intensive task handled so far has been using Shor’s algorithm to factor the number 15 on a seven-qubit system.

What prevents us from building a quantum computer today? One fundamental problem is that qubits’ states are extraordinarily fragile. Ideally, a qubit would remain exactly as we leave it, isolated from all unwanted or unknown influences (known as “decoherence”) which might alter its state and thus its information.

But in the world described by modern physics, nothing is truly in isolation; temporally sustaining a specific quantum state in a qubit poses a formidable engineering problem. In the context of quantum computation, decoherence is a source of noise, threatening the memory of our device.

A second major problem is the difficulty of wiring qubits together into a functional architecture. To function as parts of a computer, qubits must interact in a prescribed way; we must be able to put a qubit in a specific state, manipulate it as we wish, and ultimately retrieve the final state.

Progress on building a quantum computer must overcome the tension between these two requirements: we want our qubits isolated from any interaction to preserve their information, but want them to interact very strongly in specific ways with other parts of the computer.

As the size of our computer increases, there are increasingly many outlets by which information can be leaked to the environment in uncontrolled ways. This quandary of assembling qubits that can work together is called the problem of scalability.

For qubits based on manmade devices the major obstacle is maintaining coherence— sustaining the probabilistic superposition of a qubit’s state. It is easier to craft a means to connect qubits if you’ve built them yourself.

For qubits in nature, (e.g. a particle’s spin) the opposite is true. Schoelkopf explains these qubits “are great because they have very long lifetimes and you can do very good quantum control. The problem is, it’s very hard to wire them up, to string them together in a complicated arrangement.”

Current research trends lean toward solid-state devices such as superconducting circuits. Experts agree that computation with “god-given” qubits probably won’t be able to breach the scale of 7-8 qubit systems that have been built thus far. Devoret notes, however, that they could perhaps be used even in a large-scale quantum computer as a form of permanent memory.

The recent work at Yale is a step toward constructing an integrated, many-qubit system. Prior methods of exchanging quantum information have relied on close proximity between qubits, but the photon exchange method described below could potentially couple distant qubits, crucial for a many-qubit architecture.

Quantum bits and gates basics

First, it will be useful to detail the general mechanisms by which qubits are used to store and manipulate information. The fundamental distinction between classical and quantum bits manifests itself in the space of possible states for each.

In classical bits, this space has two elements, 0 and 1. In quantum mechanics, for each possible state there is some complex number whose magnitude squared is the probability of that state.

This can be a continuous complex “wavefunction” (e.g. for position), or a discrete set of numbers (e.g. two numbers for a qubit). The general complex number is one real number plus another real number times i, so one complex number can be represented as a point on the space of real numbers, i.e. the plane.

It is convenient to specify complex numbers by their distance from the origin (magnitude), and their angle in the plane (phase). The space of two complex numbers (two planes) has four real dimensions, but for qubits their squared magnitudes must add to one to give a probability distribution (which removes one dimension).

Rotating both numbers in the complex plane gives an identical single-qubit state where only the relative magnitude and relative phase matter, leaving two continuous real dimensions.

This space, mathematically defined as the set of two complex numbers up to a complex multiple, is isomorphic to the space of points on the surface of a figure known to physicists as the Bloch sphere (Fig. 1).

Fig. 1: A qubit’s state is specified by a point on a sphere. The relative phase of the qubit’s weights is φ. The magnitude of the “1” state is cosθ, the magnitude of “0” is sinθ, and the corresponding probabilities are cos2θ and sin2θ. At the north pole the qubit holds 1, at the south pole it holds 0, and on the equator it’s 50/50.

The relative magnitude of the two probabilities corresponds to latitude on the Bloch sphere; the relative phase is given by the longitudinal angle.

For spin-based qubits, whose two states have the particle’s magnetic moment aligned and anti-aligned with an external field in z direction, position on the Bloch sphere corresponds to the mean measured direction of the particle’s actual angular momentum.

In both classical and quantum computation, bits are manipulated by sending them through “gates.” For instance, classical NOT and OR gates take one and two bits, respectively, and output one.

Quantum gates, however, always input and output the same number of bits and are always reversible: one can deduce the input from the output. This is because according to quantum mechanics, present wavefunctions are mapped to future wavefunctions by a class of mappings called “unitary transformations,” and these are always invertible.

A one-bit quantum gate corresponds simply to some rotation of the qubit’s position on the Bloch sphere; rotating one or both of the qubit’s weights in the complex plane corresponds to linear, normalization-preserving transformations, and these correspond to changes in Bloch sphere longitude (or latitude). Mathematically, this means varying their magnitudes while keeping the sum of squared values equal to one, like the sine and cosine functions.

Changes in longitude (relative phase of the weights), which leave the qubit with the same probability of holding 0 or 1, are easy to obtain. We both predict and observe that the state of any two-level quantum system constantly rotates longitudinally on the Bloch sphere as it evolves in time.

For a charged particle with spin in a magnetic field, Bloch sphere axes are real spatial axes and this rotation is the phenomenon of spin precession, crucial in nuclear magnetic resonance; but remarkably, the result follows directly from the Schrödinger equation and applies to all two-level quantum systems, which come in a staggering range of physical varieties.

But to manipulate a qubit in a computationally useful manner, we must also alter its probabilities. We can accomplish this by perturbing the system with a force that oscillates at the system’s resonant frequency, proportional to the energy difference between the two levels.

Driving at resonant frequency causes the latitude of the qubit to sinusoidally oscillate as it absorbs and emits energy. Its position on the Bloch sphere spirals about, spinning longitudinally while bouncing up and down—both a theoretical and experimental result.

By controlling the temporal duration or strength of the driving, we can control the rotation angle to manipulate the probabilities that the qubit will be in either state.

In a quantum computer, ideally, many qubits would act in concert. The first step toward this coordination is the two-bit quantum gate, an operation performed on two qubits such that the final state of one depends on the state of the other.

There are countless conceivable two-bit gates, but the one which has received the most attention among physicists trying to build a quantum computer is the CNOT gate. In this, the first bit is flipped if the second bit is 1, but left the same if the second bit is 0, or vice versa.

Another novel feature of quantum computation is the universality of the CNOT gate. CNOTs can be combined to produce any other possible gate, and thus in principle execute any terminating computation. To construct a quantum CNOT, two qubits must be coupled so that the frequency of one depends on the state of the other.

After this, we can sinusoidally drive the first qubit at a frequency that is only just right if the second bit is in a certain state. Since a qubit’s natural frequency is proportional to the energy gap between its two states, we must arrange this energy to be influenced by the state of the second bit.

The mechanism by which this is accomplished depends on the type of qubit. For spins, coupling of magnetic moment can be used; for charge-based superconducting circuit qubits in proximity, the charge distribution across one will alter the energy required to move a charge across the other.

The quantum bus

Coupling mechanisms developed so far have relied on the physical proximity of the qubits. The primary purpose of the “quantum bus” is to potentiate longdistance coupling of quantum states by sending information-carrying photons between qubits.

In two 2007 Nature articles, the Yale researchers described their progress toward such a device. The first study demonstrated a mechanism to transfer information from a charge-based qubit to a moving system whose two states are the existence or absence of a photon.

This was the first step toward a mechanism for long-distance quantum information transfer—taking the information in a qubit with fixed position and putting it in a traveling photon (or “flying qubit”).

However, a long-distance two-bit gate also requires the reverse task: putting this information into another charge-based qubit. This was demonstrated in the second paper: quantum information sent back and forth between two charge based qubits with a resonating photon (a standing electromagnetic wave) bouncing between them.

The quantum bus is the union of several carefully engineered components, each developed independently yet working together. The backbone and transport mechanism of the device is an electromagnetic chamber known as a “cavity,” which houses light waves with amplitude reaching to the walls of the cavity (Fig. 2).

Fig 2: A photo of the very small experimental setup. To the right is a penny.

This is manifested as a stretch of winding wire about 0.01 mm wide and 12 mm long. When current passes through normal wires, the ripple in the electromagnetic field extends outside of the wire. In a coaxial cable (like those that bring information to televisions), this electromagnetic signal is confined within a tube of metal surrounding a central wire.

The tiny cavity in the quantum bus is a 2-D version of the coaxial cable called a coplanar waveguide: electromagnetic waves are trapped within a long, skinny winding box of metal housing a central wire.

The central wire is clipped on either end to give two breaks in the wire; these act as mirrors for the electromagnetic waves so they bounce back and forth as standing waves. Note that in the first experiment, the properties of the two gaps are offset so that the photon exits traveling out one side instead of resonating.

A photon resonating or traveling in the cavity corresponds to electrons’ motion in the wires. According to Girvin, “You can either think of that as charges sloshing back and forth in the wires, or you can think of it as photons traveling down through the empty space between the wires.”

Normally, when electrons move in a wire, they bump into atoms in the surrounding metal, transferring energy which is ultimately radiated as heat and lost to the system.

This corresponds to the resistance of the wire, and is why a circuit must be continually fed with energy to maintain a current. However, electrons in the quantum bus cavity are “superconducting”; that is, they travel in pairs and do not lose energy by crashing into atoms.

Circumventing this energy loss is crucial for maintaining coherence of quantum information held in the circuit. All uncontrolled routes of energy loss are sources of decoherence, so much of the effort in building a quantum computer tries to find and suppress these forms of “relaxation.”

Generally speaking, irreversible computations must dissipate energy as heat, adding entropy to the universe. However, the converse is also true: if quantum computation operations are reversible, they must not dissipate energy. Also, superconductivity is a fundamentally quantum phenomenon, so small superconducting circuits are well-suited to house a system of qubits.

Because metals typically superconduct only at very low temperatures, the quantum bus is cooled to 20 milliKelvin. Low temperature is important for sustaining quantum information: at high temperatures a qubit will be more likely to randomly receive energy from its surroundings and end up in the excited state.

How is a superconducting circuit used as a system of qubits? In small enough circuits, degrees of freedom such as the charge across a capacitor must be treated as quantum (probabilistic) variables.

One superconducting circuit element that is intrinsically quantum mechanical is the Josephson junction, a thin (~10-atom) barrier that makes a break in a superconducting wire like a capacitor, but thin enough that electrons can tunnel through and jump to the other side, allowing current to flow.

Yale researchers have developed a Josephson junction-based qubit called the transmon that is designed to be easily integrated into a larger circuit like the aforementioned cavity. In this design, two Josephson junctions connect either side of a small loop of metal onto a much longer wire.

The small loop, called an “island,” can receive any number of electrons from the enormous amount in the wire via tunneling; each electron on the island beyond the equilibrium charge distribution gives a quantized higher energy state.

We can treat the lowest two states of this system as a qubit if two conditions are met. (1) The temperature is sufficiently low that random excitations are unlikely to occur. (2) The energy gap between the lowest two levels differs from the gaps between other pairs of levels so that driving at the proper frequency will cause transitions only between the lowest two states.

In the quantum bus experiments, the “longer wire” for these qubits was the 2D coaxial-esque wire described above. One or two qubits sit inside the cavity, coupled via electromagnetic interaction so that the existence or absence of a resonant photon in the cavity depends on their energy state.

In the first study, a single transmon qubit was placed in the cavity. An input pulse— an electromagnetic wave shaped like a normal distribution, is sent in on one end of the waveguide to excite the qubit, which then releases the energy as a photon exiting on the other end.

The bell-curve shape is given by a sum of many photons of different frequencies, including the right frequency to excite the transmon. Energy is absorbed at only this resonant frequency, and the rest of the wave passes through.

The oscillation rotates the qubit’s state, changing its latitude, and the angle of this rotation increases linearly with the strength of the input pulse. Maximum latitude corresponds to the absorption of a single full packet of electromagnetic energy (a photon), and lower latitudes correspond to some smaller probability of absorption.

Shortly after absorbing the photon (but after the rest of the input pulse has left), the qubit emits a probabilistic signal, a superposition of the existence and absence of an emitted photon, with precisely the same quantum information as the stationary qubit had before.

This measurement was made for a wide range of input pulse strengths, and the latitude of the output signals varied like the sine of pulse strength.

Besides demonstrating quantum information transfer from a stationary circuit element to a moving photon, it is remarkable that this system is able to generate a single, discrete photon as output. Electromagnetic devices typically emit many photons at once, e.g. a cell phone emits about 1023 microwave photons per second.

Single photon signals are necessary to encode individual quantum states; the low temperature and energy of the system contribute crucially to the capability of generating single microwave photons, which have extremely low energy compared to other parts of typical circuits.

In the second study, the cavity houses two qubits with about 10 mm of waveguide wire in between. An electromagnetic input pulse excites one of the qubits, putting it in the pure excited “1” state. The “mirror” gaps are adjusted so that the photon emitted by this first qubit doesn’t exit through the other side; it remains in the cavity as a resonant standing wave.

Therefore, the photon bounces between the two transmon qubits. First, one transmon is excited, the other is relaxed, and no photon is present. Next, a photon exists in the cavity, and the two transmons are in an “entangled” (probabilistically dependent) state known as an EPR (Einstein-Podolsky- Rosen) pair, with the energy of one dependent on the energy of the other.

Finally, the second transmon takes up the probabilistic packet of energy and is excited, while the resonant photon and first transmon relax—and the process then repeats (probably about twenty times, according to Girvin, although the exact number is not known).

This two-bit quantum gate transformation is called an “iSWAP.” Like the CNOT, it is a universal gate. The photon is “virtual”; it carries more energy than would be permitted by energy conservation, but vanishes after a several nanosecond lifetime, so that energy is conserved in the long run.

Girvin describes the role of this virtual photon in transmitting the energy swap between qubits: “The first qubit makes the photon, then realizes, ‘Oh, I can’t conserve energy, I have to stop,’ and the energy either has to go back or jump onto the other qubit. The photon winks into existence and then disappears before you could tell that it violated energy conservation.”

Schematic of a quantum cavity bus chip

The road to a quantum computer

The photon transfer mechanism implemented in the Yale researchers’ quantum bus is an early but important piece of progress toward the vision of a large-scale quantum computer. Schoelkopf described the studies as the first step in making the principles of quantum computing useful.

Photons can travel on microwave lines for up to ten kilometers, and more qubits could conceivably be coupled to the bus’s cavity, opening the potential for longdistance information transfer between many bits.

The professors stress that their inventions are works in progress. For now, the bus can only swap energy between qubits a few times, and the number of swaps isn’t precisely known or controllable.

Given our current progress, the ultimate goal of a quantum replacement for a desktop seems quite distant. Considering the tremendous difficulty in controlling one or two-qubit systems and the formidable intricacy of the theoretical analysis required, the hope of performing computations over thousands or millions of integrated qubits might seem vain.

In any case, a large body of research toward building a quantum computer has emerged since theoretical work in the ‘80s demonstrated its remarkable computational potential. At the very least, many physicists hope and believe that quantum computation will prove useful.

The creation of a full-fledged quantum computer, a delicately controlled bulk system, would represent a tremendous technological feat. The development of a means to transfer quantum information on photons has brought us one step closer to this vision.

About the Author
Ben Deen is a senior in Trumbull College double-majoring in physics and cognitive science.

Further Reading

  • Houck et al. (2007). Generating single microwave photons in a circuit. Nature 449, p. 328-331.
  • Majer et al. (2007). Coupling superconducting qubits via a cavity bus. Nature 449, p. 443-447.
Ben Deen
By Ben Deen February 26, 2009 15:18