Semiconductor quantum dots take first steps toward spin-based quantum computation

Of the obstacles to building a quantum computer, two remain especially tough to surmount: scaling up from a few-qubit logic gate to a many-qubit device and sustaining the qubits’ coherence long enough to do a worthwhile and accurate calculation.

In principle, both obstacles could be tackled by basing a quantum computer on electron spins in semiconductors. Intel, Samsung, and other chipmakers already integrate tens of millions of logic gates on dime-sized wafers of silicon. And the up-or-down spin of an electron, a natural qubit, couples to the surrounding lattice with coherence-preserving weakness.

But weak coupling has the defect of its virtue. In quantum computation one needs to manipulate individual qubits. Grasping a particular spin by its puny magnetic field, setting its initial state, flipping it on cue, and reading its final state are beyond today’s technology.

Fortunately, the electron’s charge provides another, firmer handle. Eight years ago, Daniel Loss of the University of Basel in Switzerland and David DiVincenzo of IBM’s T. J. Watson Research Center in Yorktown Heights, New York, proposed a concept they called spin–charge conversion. ¹ Single spins are put into semiconductor quantum dots and controlled with voltage pulses. Thanks to the Pauli exclusion principle and Zeeman splitting, up spins and down spins respond differently to the same electric field.

That way, spins can be moved and entangled, but spin-based quantum computation also requires rotating individual spins. Radio waves and strong magnetic fields could do that, just as in NMR, but the radiation required would roast the dots.

In 2002, Jeremy Levy of the University of Pittsburgh devised a way of doing spin–charge conversion at lower fields and frequencies. Instead of individual spins, he proposed using spin pairs. ² The pair’s singlet state would form the qubit’s 0; one of the triplets would form the qubit’s 1; and the two states’ entangled combinations would form the Hilbert space where the unitary operations that underlie quantum computation take place.

Using two spins per qubit comes at a cost, however. Forestalling a qubit’s decoherence requires protecting the alignment of two spins simultaneously at different locations. In that case, the qubit is vulnerable not only to whatever sources of decoherence lurk in the lattice but also to the sources’ spatial variation. How long can a spin-paired qubit survive?

A collaboration based at Harvard University now has the answer. ³ By measuring the effect of nanosecond voltage pulses on a pair of gallium arsenide quantum dots, Harvard’s Jason Petta, who’s a postdoc in Charles Marcus’s lab, and his colleagues have determined that a singlet state will decohere in about 10 ns.

That time scale, which arises from the interaction of the electron spins with the nuclear spins of gallium and arsenic, is too short for computation. But remarkably, using a technique borrowed from NMR, the Harvard collaboration can stave off decoherence for 1 µs. That still seems short. But if a further factor of 10 could be obtained, and if the qubit could be controlled with the gigahertz clock speeds of today’s fastest microprocessors, then coherence would last long enough to meet a key benchmark: the ability to perform 10 000 operations.

Double dots

At first, the Harvard collaboration set out simply to measure how long a singlet state, separated in two dots, survives before decoherence destroys it. Figure 1 shows the experiment. Two gates, L and R, control the occupancy of the dots, while the gate labeled T controls tunneling between the dots. Tuning the voltage difference ε between the left and right gates puts two electrons in the right dot or one each in the right and left dots. A quantum point contact (QPC) measures the charge in the right dot.

Figure 2 outlines the plan of action. At first, ε is set to load both electrons into the right dot, a charge state denoted by (0,2). The temperature, about 135 mK, is low enough that electrons sit in the singlet state, denoted by (0,2)S, and can’t reach the higher-energy triplet states. Lowering ε with a few-nanosecond voltage pulse lets one of the electrons tunnel into the left dot. Because tunneling preserves spin, the singlet initially survives. After an adjustable evolution time in the range of 0–50 ns, ε is raised again. If, at the end of the evolution time, the electrons are in the singlet state, the left electron can tunnel back into the right dot. But if the singlet state decoheres, the act of trying to pull the left electron back can collapse the state into a triplet, stranding the electron.

Which of the triplets the state can end up in depends, as figure 3 shows, on ε and on whether an external magnetic field B splits the triplets T₋, T₀, and T₊. When ε ≪ 0, the separated singlet S is degenerate with all three triplets (B = 0) or with just T₀ (B > 0). At finite B, and at a specific B-dependent value of ε, S is degenerate only with T₊.

The exquisitely sensitive QPC can read out the number of electrons on the right dot, but it can’t keep up with the nanosecond time scale of the pulses. To overcome that limitation, the measurement phase is set to last 10 times longer than the loading, separation, and evolution phases. Repeating the experiment 100 000 times for each value of evolution time ensures the QPC is overwhelmingly likely to record the mean charge state at the end of the evolution phase.

From the final charge state as a function of evolution time, the Harvard collaboration derived P _S, the probability of the final state being a singlet. At large values of evolution time, when decoherence has set in and all the degenerate states are equally likely, P _S should saturate at 1/2 when B = 0 and 1/3 when B = 100 mT. Those final values of P _S (plus an experimental offset of 0.2) are reached on a time scale of 10 ns.

Spin echo

The Harvard quantum dots are about 50 nm in diameter and contain about a million nuclear spins (gallium and arsenic each have a spin of 3/2), whose orientations fluctuate randomly on a millisecond time scale. During each measurement of P _S, their magnetic field B _nuc is pretty much static. To flip an electron’s spin in 10 ns, the nuclear spin field would have to be about 2 mT, which is consistent with previous observations. ⁴

Because the nuclear spins fluctuate, it’s unlikely that B _nuc is the same in each dot. The difference in fields, ΔB _nuc, creates an additional but small hyperfine splitting between the states |↓ ↑ 〉 and |↑ ↓ 〉. That splitting can be exploited both to prolong coherence and to rotate a qubit. Here’s how.

The eigenstates of spin exchange, S and T₀, can be written in terms of the eigenstates of the hyperfine splitting, |↑ ↓ 〉 and |↓ ↑ 〉, as

$$\begin{array}{c}\text{S}=\left(|\downarrow \uparrow 〉-|\uparrow \downarrow 〉\right)/\surd 2\hspace{0.17em}\text{and}\\ {\text{T}}_{\text{0}}=\left(|\downarrow \uparrow 〉+|\uparrow \downarrow 〉\right)/\surd 2\hspace{0.17em}.\end{array}$$

As S decoheres, the changing mix of eigenstates appears in the Bloch sphere as a dephasing in a single plane. That dephasing can be canceled by applying a technique borrowed from NMR called spin echo.

Spin echo works by inverting the electron’s spin configuration and allowing the equal but now opposite effect of ΔB _nuc to cancel the initial dephasing. Figure 4 illustrates the scheme.

The dephasing and rephasing both occur when the (1,1) charge state is strongly favored—that is, when ε ≪ 0. Inversion, which corresponds to a rotation in the Bloch sphere, occurs when ε is raised just long enough to let the electrons tunnel between dots and complete an odd number of Heisenberg spin exchanges.

That exchange time, τ _E, is given by J(ε)τ _E/ħ = π, 3π, 5π, …, where J(ε) is the energy cost of doing the exchange. In figure 3, J(ε) corresponds to the difference in height between the S and T₀ curves. The Harvard collaboration derived J(ε) from the behavior of P _S at the degeneracy between S and T₊.

Figure 4 illustrates how spin echo can prolong the qubit’s coherence. When the dephasing and rephasing times are the same and τ _E is appropriately tuned, P _S decays with a best-fitting timescale of 1.2 µs. When the dephasing and rephasing times are different, P _S decays on a best-fitting timescale of 9 ns, which is consistent with not doing any spin echo, as before.

What could be responsible for the 1-µm decoherence time? One possibility is the virtual flipping of the nuclear spins. The electron’s wavefunction spreads over many lattice sites. Even if the mean value of B _nuc is steady, the random, virtual flip-flopping of nuclear spins creates a torque on the electron’s spin whose strength depends on where each flip occurs in relation to the peak of the electron’s wavefunction. To pin down the cause, Petta and coworkers are already investigating its dependence on B and other quantities.

Toward a quantum computer

To quantum compute, one needs, among other things, to rotate one qubit state into another. The nuclear spins can rotate S into T₀ and vice versa, but not by a predictable amount. Given that exchange can control the mix of |↓ ↑ 〉 and |↑ ↓ 〉, can those states be used as a possible basis for a qubit’s 0 and 1?

Yes, according to the Harvard collaboration, provided the electron pair, just before rotation, is in |↓ ↑ 〉 or |↑ ↓ 〉, rather than S or T₀. Consider the case when |↓ ↑ 〉 happens to be lower in energy than |↑ ↓ 〉. If the spins are separated adiabatically—that is, slowly enough to let B _nuc torque the spins—they’ll settle into their ground state |↓ ↑ 〉. From there, exchange can be turned on to make one or more complete inversions, thereby converting |↓ ↑ 〉 to |↑ ↓ 〉. Rejoining the spins adiabatically turns |↑ ↓ 〉 into T₀; plotting the singlet survivability P _S as a function of τ _E results in oscillations whose minima occur when J(ε)τ _E/ħ = π, 3π, 5π, and so on—which is what the Harvard collaboration observed.

What’s more, by using the same scheme but with half the value of exchange (that is, π/2, 3π/2, 5π/2, …), the Harvard collaboration could perform the so-called square-root-of swap operation. That operation forms the basis for the XOR logic gate in Loss and DiVincenzo’s 1998 proposal.

Recruiting the hyperfine splitting to rotate qubits is far from ideal, however, because it depends on the chance and unpredictable value of ΔB _nuc. Establishing a constant ΔB _nuc would be better. Also needed for quantum computation are two-qubit operations and the ability to initialize and read out single qubits. (Averaging 100 000 measurements with a QPC doesn’t count.)

Whether those and other obstacles will ever be overcome remains uncertain. But progress has been surprisingly quick. Says Sankar Das Sarma of the University of Maryland in College Park, “I didn’t think anyone would be able to manipulate and decouple single spins until 2010!”