## Entangling Qubits

The basic building block of a quantum computer, a qubit, has been realized in many physical settings, each of which has its advantages and drawbacks. Solid-state spin qubits interact weakly with their environment and each other, leading not only to long coherence times but also to difficulties in performing multiqubit operations. **Shulman et al.** (p. 202) used a double quantum dot to produce a singlet-triplet qubit, where the two quantum states available are a singlet and a triplet formed by two spin-1/2 electrons. Two such qubits are then entangled by electrical gating, which affects the charge configuration of one qubit and that, in turn, influences the electric field experienced by the other. This type of two-qubit entanglement is essential for further development of quantum computing in these systems.

## Abstract

Quantum computers have the potential to solve certain problems faster than classical computers. To exploit their power, it is necessary to perform interqubit operations and generate entangled states. Spin qubits are a promising candidate for implementing a quantum processor because of their potential for scalability and miniaturization. However, their weak interactions with the environment, which lead to their long coherence times, make interqubit operations challenging. We performed a controlled two-qubit operation between singlet-triplet qubits using a dynamically decoupled sequence that maintains the two-qubit coupling while decoupling each qubit from its fluctuating environment. Using state tomography, we measured the full density matrix of the system and determined the concurrence and the fidelity of the generated state, providing proof of entanglement.

Singlet-triplet (*S*-*T*_{0}) qubits, a particular realization of spin qubits (*1*–*7*), store quantum information in the joint spin state of two electrons (*8*–*10*). The basis states for the *S*-*T*_{0} qubit can be constructed from the eigenstates of a single electron spin, |↑〉 and |↓〉. We chose |*S*〉 = (1/√2)(|↑↓〉 – |↓↑〉) and |*T*_{0}〉 = (1/√2)(|↑↓〉 + |↓↑〉) because these states are insensitive to uniform fluctuations in the magnetic field. The qubit can then be described as a two-level system with a representation on the so-called Bloch sphere (Fig. 1A). Universal quantum control is achieved using two physically distinct operations that drive rotations around the *x* and *z* axes of the Bloch sphere (*11*). Rotations around the *z* axis are driven by the exchange splitting, *J*, between |*S*〉 and |*T*_{0}〉, and rotations around the *x* axis are driven by a magnetic field gradient, ∆*B _{z}*, between the electrons.

We implemented the *S*-*T*_{0} qubit by confining two electrons to a double quantum dot (QD) in a two-dimensional electron gas (2DEG) located 91 nm below the surface of a GaAs-AlGaAs heterostructure. We deposited local top gates using standard electron beam lithography techniques to locally deplete the 2DEG and form the QDs. We operated between the states (0,2) and (1,1), where (*n _{L}*,

*n*) describes the state with

_{R}*n*(

_{L}*n*) electrons in the left (right) QD. The |

_{R}*S*〉 and |

*T*

_{0}〉 states, the logical subspace for the qubit, are isolated by applying an external magnetic field of 700 mT in the plane of the device such that the Zeeman splitting makes

*T*

_{+ }= |↑↑〉 and

*T*

_{− }= |↓↓〉 energetically inaccessible. The exchange splitting,

*J*, is a function of the difference in energy, ε, between the levels of the left and right QDs. Pulsed DC electric fields rapidly change ε, allowing us to switch

*J*on, which drives rotations around the

*z*axis. When

*J*is off, the qubit precesses around the

*x*axis due to a fixed ∆

*B*, which is stabilized to ∆

_{z}*B*/2π = 30 MHz by operating the qubit as a feedback loop between iterations of the experiment (

_{z}*12*). Dephasing of the qubit rotations reflects fluctuations in the magnitude of the two control axes,

*J*and ∆

*B*, caused by electrical noise and variation in the magnetic field gradient, respectively. The qubit is rapidly (<50 ns) initialized in |

*S*〉 by exchanging an electron with the nearby Fermi sea of the leads of the QD, by tuning the QD potentials so that only |

*S*〉 lies below the Fermi energy. The qubit state is read out using standard Pauli blockade techniques, where ε is quickly tuned to the regime where S occupies (0,2) and

*T*

_{0}occupies (1,1), allowing the qubit state to be determined by the proximal charge sensor. The charge state of the qubit is rapidly determined (∼1 μs) using standard radio frequency techniques (

*13*,

*14*) on an adjacent sensing QD.

To make use of the power of quantum information processing, it is necessary to perform two qubit operations in which the state of one qubit is conditioned on the state of the other (*15*). To investigate two-qubit operations, we fabricated two adjacent *S*-*T*_{0} qubits such that they are capacitively coupled, but tunneling between them is suppressed (Fig. 1B). A charge-sensing QD next to each qubit allows for simultaneous and independent projective measurement of each qubit (see supplementary materials). We used the electrostatic coupling between the qubits to generate the two-qubit operation (*16*). When *J* is nonzero, the |*S*〉 and |*T*_{0}〉 states have different charge configurations in the two QDs because of the Pauli exclusion principle (Fig. 1C). This charge difference, which is a function of ε, causes the |*S*〉 and |*T*_{0}〉 states in one qubit to impose different electric fields on the other qubit. As *J* is a function of the electric field, the change in electric field imposed by the first qubit causes a shift in the precession frequency of the second qubit. In this way, the state of the second qubit may be conditioned on the state of the first qubit. More precisely, when a single qubit evolves under exchange, there exists a state-dependent dipole moment, , between |*S*〉 and |*T*_{0}〉, resulting from their difference in charge occupation of the QDs. Therefore, when simultaneously evolving both qubits under exchange, they experience a capacitively mediated, dipole-dipole coupling that can generate an entangled state. The two-qubit Hamiltonian is therefore given by:(1)where σ_{x}_{,}_{y}_{,}* _{z}* are the Pauli matrices,

*I*is the identity operator, ∆

*B*

_{z}_{,}

*, and*

_{i}*J*are the magnetic field gradients and the exchange splittings (

_{i}*i*= 1,2 respectively for the two qubits), and

*J*

_{12}is the two-qubit coupling, which is proportional to the product of the dipole moments in each qubit. For a two-level system with constant tunnel coupling, the dipole moment scales as ∝ ∂

*J*/∂ ϵ

_{i}*. Empirically, we find that for experimentally relevant values of*

_{i}*J*, ∂

_{i}*J*/∂ ϵ

_{i}*∝*

_{i}*J*(ε), so that

_{i}*J*

_{12}∝

*J*

_{1}

*J*

_{2}. As with the single qubit operations, this two-qubit operation requires only pulsed DC electric fields.

In principle, evolving both qubits under exchange produces an entangling gate. However, the time to produce this maximally entangled state exceeds the inhomogeneously broadened coherence times of each individual qubit, rendering this simple implementation of the two-qubit gate ineffective. To mitigate this, we used a dynamically decoupled entangling sequence (*17*, *18*)(Fig. 1D). In this sequence, each qubit is prepared in |*S*〉 and is then rotated by π/2 around the *x* axis (*J _{i }*= 0, ∆

*B*

_{z}_{,}

*/2π ≈ 30 MHz) to prepare a state in the*

_{i}*x*-

*y*plane. The two qubits are subsequently both evolved under a large exchange splitting (

*J*

_{1}/2π ≈ 280 MHZ,

*J*

_{2}/2π ≈ 320 MHz >> ∆

*B*) for a time τ/2, during which the qubits begin to entangle and disentangle. A π pulse around the

_{z}*x*axis (∆

*B*) is then applied simultaneously to both qubits, after which the qubits are again allowed to exchange for a time τ/2. This Hahn echo–like sequence (

_{z}*19*) removes the dephasing effect of noise that is low frequency compared to 1/τ, and the π pulses preserve the sign of the two-qubit interaction. The resulting operation produces a controlled phase (CPhase) gate, which, in a basis of {|

*SS*〉,|

*T*

_{0}

*S*〉,|

*ST*

_{0}〉,|

*T*

_{0}

*T*

_{0}〉}, is an operation described by a matrix

*diag*(

*e*

^{−}

^{i}^{θ/2},1,1,

*e*

^{−}

^{i}^{θ/2}). For τ = τ

*= , the resulting state is a maximally entangled generalized Bell state |Ψ*

_{ent }*〉 =*

_{ent}*e*

^{i}^{π(}

^{I⊗σy+}

^{σy⊗I}

^{)/8}|Ψ

_{−}〉, which differs from the Bell state |Ψ

_{−}〉 = (1/√2)(|

*SS*〉 − |

*T*

_{0}

*T*

_{0}〉) by single-qubit rotations.

To characterize our two-qubit gate and verify that we produced an entangled state, we performed two-qubit state tomography and extracted the density matrix and appropriate entanglement measures. The tomographic procedure is carefully calibrated with minimal assumptions to avoid adding spurious correlations to the data that may artificially increase the measured degree of entanglement (fig. S4). We chose the Pauli set representation of the density matrix (*15*, *20*, *21*), where we measured and plotted the 16 two-qubit correlators 〈*ij*〉 = 〈σ* _{i}*σ

*〉 where σ*

_{j}*are the Pauli matrices and*

_{i}*i*,

*j*∈ {

*I*,

*X*,

*Y*,

*Z*}. As a first measure of entanglement, we evaluated the concurrence (

*22*) (Fig. 2A),

*C*(ρ) = max{0, λ

_{4 }− λ

_{3 }− λ

_{2 }− λ

_{1}} for different τ, where ρ is the experimentally measured density matrix and λ

*are the eigenvalues, sorted from smallest to largest, of the matrix , and = (σ*

_{i}*⊗σ*

_{y}*)ρ*(σ*

_{y}*⊗σ*

_{y}*), and ρ* is the complex conjugate of ρ. A positive value of the concurrence is a necessary and sufficient condition for demonstration of entanglement (*

_{y}*22*). For τ = 140 ns, we extracted a maximum concurrence of 0.44.

A positive value of the concurrence is a definitive proof of entanglement; however, it alone does not verify that the two-qubit operation produces the intended entangled state. To better characterize the generated quantum state, we evaluated another measure of entanglement, the Bell state fidelity, *F* ≡ 〈Ψ* _{ent}*|ρ|Ψ

*〉. This may be interpreted as the probability of measuring our two-qubit state in the desired |Ψ*

_{ent}*〉. Additionally, for all nonentangled states, one can show that*

_{ent}*F*≤ 0.5 (

*23*,

*24*). The Bell state fidelity takes the simple form

*F*= (1/4)

_{}× , where

_{}and

_{}are the Pauli sets of a pure target Bell state and of the experimentally measured state, respectively. For our target state |Ψ

*〉, the resulting Pauli set is given by 〈*

_{ent}*XZ*〉 = 〈

*ZX*〉 = 〈

*YY*〉 = 1, with all other elements equal to zero (Fig. 3A).

In an idealized, dephasing-free version of the experiment, as τ increases and the qubits entangle and disentangle, we expect the nonzero elements of the Pauli set for the resulting state to be(2)Dephasing due to electrical noise causes the amplitudes of the Pauli set to decay. However, the two-qubit Hamiltonian (Eq. 1) includes rapid single-qubit rotations around the S-*T*_{0} axis (*J*_{1},*J*_{2} >> *J*_{12}/2π ≈ 1 MHz) that change with τ because of imperfect pulse rise times in the experiment. These contribute additional single-qubit rotations around the *S*-*T*_{0} axis of each qubit, not accounted for in Eq. 2. We determined the angle of the single-qubit rotations by performing a least-squares analysis to find the single-qubit rotations that map the experimental data to the expected state without rotations (eq. S1), which is a modified form of Eq. 2 that accounts for dephasing. The decays due to dephasing were fit by calculating ρ(*t*) in the presence of noise on *J*_{1} and *J*_{2}, which leads to decay of certain terms in the density matrix (*25*, *26*). For the present case, where *J*_{12} << *J*_{1}, *J*_{2}, we neglected the two-qubit dephasing, which is smaller than single-qubit dephasings by a factor of *J*_{1}/*J*_{12}, *J*_{2}/*J*_{12} ≈ 300, and we extracted a separate dephasing time for each individual qubit. We removed the single-qubit rotations numerically to simplify the presentation of the data (Fig. 3E). The extracted angles exhibit a smooth monotonic behavior that is consistent with their underlying origin (fig. S5).

In the absence of dephasing, we would expect the Bell state fidelity to oscillate between 0.5 for an unentangled state and 1 for an entangled state as a function of τ. This oscillation is caused by the phase accumulated by a CPhase gate between the two qubits. However, the qubits dephase as the state becomes increasingly mixed, and the amplitude of the oscillation decays to 0.25. Indeed, the following behavior is observed (Fig. 2B): For very short τ, there is very little dephasing present, and the qubits are not entangled. As τ increases, the Bell state fidelity increases as the qubits entangle, reaching a maximum value of 0.72 at τ = 140 ns. As τ is increased further, we continue to see oscillations in the Bell state fidelity, but because of dephasing, they do not again rise above 0.5.

Figure 2C shows these oscillations in Bell state fidelity as a function of τ for several different values of *J* as ϵ is changed symmetrically in the two qubits. We see that as the value of *J* increases in the two qubits, the time required to produce a maximally entangled state, τ* _{ent}*, decreases, but the maximum attainable fidelity is approximately constant. This is consistent with the theory that

*J*

_{12}∝ ∂

*J*

_{1}/∂ϵ

_{1}· ∂

*J*

_{2}/∂ϵ

_{2}∝

*J*

_{1}·

*J*

_{2}.

To further understand the evolution of the quantum state, we focused on one value of *J* and compared the measured Pauli set to that expected from single-qubit dephasing rates and *J*_{12} (eq. S1). Figure 3A shows the Pauli set for the measured and expected quantum states for τ = 40 ns, which shows three large bars in the 〈*YI*〉, 〈*IY*〉, and 〈*YY*〉 components of the Pauli set. This is a nearly unentangled state. At τ = 140 ns, we see weight in the 〈*XZ*〉, 〈*ZX*〉, and 〈*YY*〉 components of the Pauli set (Fig. 3B), and we extracted a Bell state fidelity of 0.72, which demonstrates the production of an entangled state. For τ = τ* _{ent }*= = 160 ns (Fig. 3C), we see a similar state to τ = 140 ns, but with less weight in the single-qubit components of the Pauli set. This state corresponds to the intended CPhase of π, although the fidelity is slightly lower than at τ = 140 ns due to additional decoherence. Finally, at τ = π/

*J*

_{12}= 320 ns (Fig. 3D), where we expect the state to be unentangled, we again see large weight in the 〈

*YI*〉, 〈

*IY*〉, and 〈

*YY*〉 components of the Pauli set, although the bars are shorter than the Pauli set for τ = 40 ns because of dephasing of the qubits. We plotted the entire Pauli set as a function of time (Fig. 3E), which clearly shows the predicted oscillation (Eq. 2) between 〈

*YI*〉,〈

*IY*〉 and 〈

*XZ*〉,〈

*ZX*〉, with decays due to decoherence.

The two-qubit gate that we have demonstrated is an important step toward establishing a scalable architecture for quantum information processing in *S*-*T*_{0} qubits. Although a Bell state fidelity of 0.72 is not as high as what has been reported in other solid state implementations of qubits (*21*, *27*), there are easily implemented improvements to this two-qubit gate. State fidelity is lost to dephasing from electrical noise, and decreasing the ratio τ* _{ent}*/

*T*

_{2}

*, where*

^{echo}*T*

_{2}

*is the single-qubit coherence time with an echo pulse, is therefore paramount to generating high-fidelity Bell states. Large improvements can be made by introducing an electrostatic coupler between the two qubits (*

^{echo}*28*) to increase the two-qubit coupling (

*J*

_{12}) and reduce τ

*. We estimate that in the absence of other losses, if an electrostatic coupler were used, a Bell state with fidelity exceeding 90% could be produced. Other improvements can be made by studying the origins and properties of the charge noise that dephases the qubit and mitigating its adverse effects in order to increase*

_{ent}*T*

_{2}

*. This would allow future tests of complex quantum operations, including quantum algorithms and quantum error correction. Finally, the addition of electrostatic couplers would allow the qubits to be spacially separated and is a path toward implementing surface codes for quantum computation.*

^{echo}## Supplementary Materials

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
**Acknowledgments:**This work is supported through the U.S. Army Research Office (ARO) Precision Quantum Control and Error-Suppressing Quantum Firmware for Robust Quantum Computing and the Intelligence Advanced Research Projects Activity (IARPA) Multi-Qubit Coherent Operations (MQCO) Program. This work was partially supported by the ARO under contract W911NF-11-1-0068. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN), which is supported by the National Science Foundation under NSF award ECS-0335765. CNS is part of Harvard University. V.U. prepared the crystal; M.D.S. fabricated the sample; and M.D.S., O.E.D., H.B., S.P.H., and A.Y. carried out the experiment, analyzed the data, and wrote the paper. The authors declare no competing financial interests.