## Abstract

The von Neumann architecture for a classical computer comprises a central processing unit and a memory holding instructions and data. We demonstrate a quantum central processing unit that exchanges data with a quantum random-access memory integrated on a chip, with instructions stored on a classical computer. We test our quantum machine by executing codes that involve seven quantum elements: Two superconducting qubits coupled through a quantum bus, two quantum memories, and two zeroing registers. Two vital algorithms for quantum computing are demonstrated, the quantum Fourier transform, with 66% process fidelity, and the three-qubit Toffoli-class OR phase gate, with 98% phase fidelity. Our results, in combination especially with longer qubit coherence, illustrate a potentially viable approach to factoring numbers and implementing simple quantum error correction codes.

Quantum processors (*1*–*4*) based on nuclear magnetic resonance (*5*–*7*), trapped ions (*8*–*10*), and semiconducting devices (*11*) were used to realize Shor’s quantum factoring algorithm (*5*) and quantum error correction (*6*, *8*). The quantum operations underlying these algorithms include two-qubit gates (*2*, *3*), the quantum Fourier transform (*7*, *9*), and three-qubit Toffoli gates (*10*, *12*). In addition to a quantum processor, a second critical element for a quantum machine is a quantum memory, which has been demonstrated, for example, using optical systems to map photonic entanglement into and out of atomic ensembles (*13*).

Superconducting quantum circuits (*14*) have met a number of milestones, including demonstrations of two-qubit gates (*15*–*20*) and the advanced control of both qubit and photonic quantum states (*19*–*22*). We demonstrate a superconducting integrated circuit that combines a processor—executing the quantum Fourier transform and a three-qubit Toffoli-class OR phase gate—with a memory and a zeroing register in a single device. This combination of a quantum central processing unit (quCPU) and a quantum random-access memory (quRAM), which comprise two key elements of a classical von Neumann architecture, defines our quantum von Neumann architecture.

In our architecture (Fig. 1A), the quCPU performs one-, two-, and three-qubit gates that process quantum information, and the adjacent quRAM allows quantum information to be written, read out, and zeroed. The quCPU includes two superconducting phase qubits (*18*, *19*, *21*, *22*), Q_{1} and Q_{2}, connected through a coupling bus provided by a superconducting microwave resonator B. The quRAM comprises two superconducting resonators M_{1} and M_{2} that serve as quantum memories, as well as a pair of zeroing registers Z_{1} and Z_{2}, two-level systems that are used to dump quantum information. The chip geometry is similar to that in (*21*, *22*), with the addition of the two zeroing registers. Figure 1B shows the characterization of the device by means of swap spectroscopy (*21*).

The computational capability of our architecture is displayed in Fig. 2A, where a seven-channel quantum circuit, yielding a 128-dimensional Hilbert space, executes a prototypical algorithm. First, we create a Bell state between Q_{1} and Q_{2} using a series of π-pulse, *22*). The corresponding density matrix _{1} and M_{2} by an iSWAP pulse (step II) (*22*), leaving the qubits in their ground states |g〉, with density matrix _{1} and M_{2}, a second Bell state with density matrix

To reuse the qubits Q_{1} and Q_{2}, for example to read out the quantum information stored in the memories M_{1} and M_{2}, the second Bell state has to be dumped (*23*). This is accomplished using two zeroing gates, by bringing Q_{1} on resonance with Z_{1} and Q_{2} with Z_{2} for a zeroing time τ_{z}, corresponding to a full iSWAP (step IV). Figure 2B shows the corresponding dynamics, where each qubit, initially in the excited state |e〉, is measured in the ground state |g〉 after ≅ 30 ns. The density matrix

The ability to store entanglement in the memories, which are characterized by much longer coherence times than the qubits, is key to the quantum von Neumann architecture. We demonstrate this capability in Fig. 2, D and E, where the fidelity and concurrence metrics (*24*) of the Bell states stored in M_{1} and M_{2} are compared with those for the same states stored in Q_{1} and Q_{2}. The experiment is performed as in Fig. 2A, but eliminating steps (III) and (IV) for memory storage, and steps (II) to (V) for qubit storage. For the qubits, the storage time τ_{st} is defined as the wait time at the end of step (I), before measuring the qubit states, whereas for the resonators the wait time is that between the write and read steps. The fidelity of the qubit states decays to below 0.2 after 400 ns, whereas for the states stored in the memories it remains above 0.4 up to ≅ 1.5 μs. Most important, after only 100 ns the state stored in the qubits does not preserve any entanglement, as indicated by a zero concurrence, whereas the memories retain their entanglement for at least 1.5 μs (Fig. 2E). We expect to take advantage of our architecture in long computations, where qubit states can be protected and reused by writing them into, and reading them out of, the long-lived quRAM.

Two-qubit gates are a vital resource for the operation of the quCPU (*2*, *3*). A variety of such gates have been implemented in superconducting circuits (*15*–*20*), with some recent demonstrations of quantum algorithms (*16*, *18*). Control Z-π (CZ-π) gates are readily realizable with superconducting qubits, due to easy access to the third energy state of the qubit, effectively operating the qubit as a qutrit (*16*, *18*, *20*, *25*). However, CZ-π gates are just a subset of the more general class of CZ-ϕ gates, obtained for the special case where the phase ϕ = π. In our architecture, the full class of CZ-ϕ gates, with ϕ from ≅ 0 to π, can be generated by coupling a qutrit close to resonance with a bus resonator.

Figure 3A shows the quantum logic circuit that generates the CZ-ϕ gate (left) and a shorthand symbol for the gate (right). The logic circuit demonstrates the nontrivial case where qubits Q_{1} and Q_{2} are brought from their initial ground state to |Q_{1}Q_{2}〉 = |ee〉 by applying a π-pulse to each qubit. The excitation in Q_{2} is then transferred into bus resonator B, and Q_{1}’s |e〉 ↔ |f〉 transition is brought close to resonance with B for the time required for a 2π rotation, where the states |Q_{1}B〉 = |e1〉 and |f0〉 are detuned by a frequency _{1} acquires the phase (*26*)
_{2}.

The time-domain swaps of |Q_{1}B〉 between the states |e1〉 and |f0〉 are shown in Fig. 3B, where the solid black line indicates the detunings and corresponding interaction times used to generate any phase 0 ≲ ϕ ≤ π (ideally ϕ → 0 when _{1} for each value of the detuning *26*), as shown in Fig. 3C.

A more sophisticated version of this experiment is performed by initializing Q_{1} and Q_{2} each in the superposition state |g〉 + |e〉. We move Q_{2}’s state into B, perform a CZ-ϕ gate with _{2}, rotate Q_{1}’s resulting state by π/2 about the *y* axis, and perform a joint measurement of Q_{1} and Q_{2}. Ideally, this protocol permits the creation of two-qubit states ranging from a product state for ϕ = 0 to a maximally entangled state for ϕ = π. In the two-qubit basis set *M*_{2} = {|gg〉, |eg〉, |ge〉, |ee〉}, the general density matrix of such two-qubit states reads*24*) of two-qubit states generated using 70 values of ϕ. Figure 3E shows three examples of

The state generated using ϕ = π/2 plays a central role in the implementation of the two-qubit quantum Fourier transform. Neglecting bit-order reversal, the quantum Fourier transform can be realized by applying a Hadamard gate to Q_{2}, followed by a CZ-π/2 gate between Q_{1} and Q_{2}, and finally a Hadamard on Q_{1} (*2*, *7*, *9*), as sketched in Fig. 3F (top left). Representing the input state of the transform as |*x*〉 (position) and the output as |*p*〉 (momentum), assuming |*x*〉 ∈ *M*_{2} and the indexes *x* and *p* are integers, with *p* ∈ {0, 1, 2, 3}, the output state is *2*, *18*), which allows us to obtain the *2*, *18*) shown in Fig. 3F (bottom).

Finally, by combining the CZ-ϕ and zeroing gates, we can implement a Toffoli-class gate (*10*, *12*, *27*), the three-qubit OR phase gate. This gate, combined with single-qubit rotations, is sufficient for universal computation. A Toffoli gate is a doubly controlled quantum operation, where a unitary operation is applied to a target qubit subject to the state of two control qubits. The canonical Toffoli is a doubly controlled NOT gate; here, we consider a doubly controlled phase gate, which is equivalent through a change of basis of the target qubit. In the canonical Toffoli gate, the control gate is applied if both control qubits, Q_{1} AND Q_{2}, are in state |e〉. In our case, the control gate is applied conditionally if the controls Q_{1} OR Q_{2} are in |e〉. Additionally, we have implemented a three-qubit gate for the logical function exclusive OR (XOR), which, even though not a Toffoli-class gate, helps to understand the more complex OR gate.

The quantum logic circuits for the XOR and OR gates are drawn in Fig. 4, A and D. The control qubits are Q_{1} and Q_{2}, and the target is the bus resonator B, effectively acting as the third qubit (as only the states |0〉 and |1〉 of B are used). The XOR gate is realized as a series of two CZ-π gates between the controls and the target, and the OR gate as the series ½ CZ-π, CZ-π, and ½ CZ-π, in an M-shaped configuration.

The truth table for the XOR gate is displayed in Fig. 4B (top). The control qubits Q_{1} and Q_{2} are assumed to be in one of the states in *M*_{2}, whereas the target B is in |0〉 + |1〉. The target acquires a phase π, corresponding to a “true” result, only when the controls are in the state |Q_{1}Q_{2} 〉 = |ge〉 or |eg〉. For the other nontrivial case |Q_{1}Q_{2}〉 = |ee〉, the target acquires 0 phase, corresponding to a “false” result. This is due to the action of the two CZ-π gates, giving a global phase π when either of the controls is in |e〉 and a phase 2π (equivalent to a 0 phase) when both are in |e〉.

The truth table can be experimentally measured by performing Ramsey experiments on the target, one for each pair of control states. The experiments are realized by (i) preparing Q_{2} in the superposition state |g〉 + |e〉 by means of a π/2-pulse; (ii) moving the state from Q_{2} into B, thus creating a |0〉 + |1〉 state in B; (iii) preparing Q_{1} and Q_{2} in each possible pair of control states in *M*_{2} by means of π-pulses; (iv) performing the XOR gate; (v) zeroing Q_{2} into Z_{2} at the end of the XOR gate; (vi) moving the final target state from B into the zeroed Q_{2}; and (vii) completing the Ramsey sequence on Q_{2} with a second π/2-pulse with variable rotation axis relative to the pulse in (i). The measurement outcomes are displayed in Fig. 4B (bottom), together with the least-squares fits used to extract the phase information associated with each value of the truth table. The Ramsey fringes for the two control states |ge〉 and |eg〉 are inverted relative to the reference state |gg〉, as expected from the XOR gate truth table.

In general, given the Q_{1}-Q_{2}-B basis set *M*_{3} = {|gg0〉, |gg1〉, |ge0〉, |ge1〉, |eg0〉, |eg1〉, |ee0〉, |ee1〉}, the vector **τ**^{XOR} of the diagonal elements associated with the ideal unitary matrix of the XOR gate reads*lmn*〉 ∈ *M*_{3}. The phase ϕ_{|}_{lmn}_{〉} can be either 0, when _{|}_{lmn}_{〉}, only seven are physically independent, as the element _{|}_{lmn}_{〉} − ϕ_{|gg0〉}, with |*lmn*〉 ∈ *M*_{3} − {|gg0〉}.

In analogy to the truth table for the target B, a table with four phase differences can also be obtained for the controls Q_{1} and Q_{2}, resulting in a total of 12 phase differences. These differences can be measured by performing Ramsey experiments both on the target and the control qubits. It can be shown that from the 12 phase differences, one can obtain the seven independent phases associated with the diagonal elements *26*), thus realizing a quantum phase tomography of the Toffoli gate (*28*). Figure 4C displays the phase tomography results for our experimental implementation of the XOR gate.

The truth table associated with the M gate is reported in Fig. 4E (top), where the only difference from the XOR gate is the phase π acquired by the target B when the controls Q_{1} and Q_{2} are loaded in state |Q_{1}Q_{2}〉 = |ee〉. In this case, the action of the first ½ CZ-π gate between Q_{1} and B shelves the |1〉 state from B to the noncomputational state |f〉 in Q_{1}, where it remains until the second ½ CZ-π gate. Moving the state of Q_{1} outside the computational space during the intermediate CZ-π gate between Q_{2} and B effectively turns off the CZ-π gate (*12*, *29*). The target B thus only acquires a total phase π due to the combined action of the two ½ CZ-π gates (see Figure 4D). The experimental truth table obtained from Ramsey fringes is shown in Fig. 4E (bottom).

The vector **τ**^{M} of the diagonal elements associated with the ideal unitary matrix of the M gate is **τ**^{M} = (1, 1, 1, −1, 1, −1, 1, −1). A similar procedure as for the XOR gate allows us to obtain the quantum phase tomography of the M gate (Fig. 4F).

Quantum phase tomography makes it possible to define the phase fidelity of the XOR and M gate,_{ϕ} is the gate root-mean-square phase error, with an upper bound of π. For the XOR gate we find that *F*_{ϕ} = 0.954 ± 0.004, and for the M gate *F*_{ϕ} = 0.979 ± 0.003.

Our results provide optimism for the near-term implementation of a larger-scale quantum processor (*1*–*3*) based on superconducting circuits. Our architecture shows that proof-of-concept factorization algorithms (*2*, *3*, *5*) and simple quantum error correction codes (*2*, *3*, *6*, *8*) might be achievable using this approach.

## Supporting Online Materials

www.sciencemag.org/cgi/content/full/science.1208517/DC1

Materials and Methods

Figs. S1 to S12

Tables S1 to S3

References

## References and Notes

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- ↵ A full gate characterization by quantum process tomography was not possible because we could only simultaneously measure two qubits, with the resonator acting as the third qubit.
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**Acknowledgments:**This work was supported by Intelligence Advanced Research Projects Activity (IARPA) under ARO award W911NF-08-1-0336 and under Army Research Office (ARO) award W911NF-09-1-0375. M. M. acknowledges support from an Elings Postdoctoral Fellowship. Devices were made at the University of California Santa Barbara Nanofabrication Facility, a part of the NSF-funded National Nanotechnology Infrastructure Network. The authors thank A. G. Fowler for useful comments on scalability, and M. H. Devoret and R. J. Schoelkopf for discussions on Toffoli gates. M.M. performed the experiments and analyzed the data. M.M. and H.W. fabricated the sample. T.Y., H.W., and Y.Y. helped with the Fourier transform, and M.N. with three-qubit gates. M.M., A.N.C., and J.M.M. conceived the experiment and cowrote the manuscript.