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 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. 1-4 based on nuclear magnetic resonance [5][6][7] , trapped ions [8][9][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 threequbit 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, e.g., 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 5,[15] [16] [17] 19, 20 and the advanced control of both qubit and photonic quantum states 7, 19, 20, 22 . We demonstrate a superconducting integrated circuit that combines a processor, executing the quantum Fourier transform and a three-qubit Toffoli-class OR 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 5,7, 19, 22 The quRAM (magenta boxes) comprises two memories M 1 and M 2 and two zeroing registers Z 1 and Z 2 . The horizontal dotted lines indicate connections between computational elements. The vertical direction represents frequency, where the memory and zeroing registers are fixed in frequency, while the qubit transition frequencies can be tuned via z-pulses (grey dashed double arrows). (B) Swap spectroscopy 7 for Q 1 (left) and Q 2 (right): Qubit excited state |e probability P e (color scale) vs. z-pulse amplitude (vertical axis) and delay time ∆τ (horizontal axis), after exciting the qubit with a π-pulse. At zero z-pulse amplitude the qubits are at their idle points, where they have an energy relaxation time T rel 400 ns. A separate Ramsey experiment yields the qubits' dephasing time T deph 200 ns. By tuning the z-pulse amplitude, the qubit transition frequencies f Q 1 and f Q 2 can be varied between 5.5 and 8 GHz. For z-pulse amplitudes indicated by B and M 1 for Q 1 , and by B and M 2 for Q 2 , the "chevron pattern" of a qubit-resonator interaction is observed 7 . The transition frequencies of B, M 1 , and M 2 are f B = 6.82 GHz, f M 1 = 6.29 GHz, and f M 2 = 6.34 GHz, respectively. From the chevron oscillation we obtain the qubit-resonator coupling strengths, which for both the resonator bus and the memories are 20 MHz (splitting) for the |g ↔ |e qubit transition, and ≈ √ 2 faster for the |e ↔ |f transition (|g , |e , and |f are the three lowest qubit states) 22 . For all resonators T rel 4 µs. Swap spectroscopy also reveals that the qubits interact with several modes associated with spurious two-level systems. Two of them, Z 1 and Z 2 , are used as zeroing registers. Their transition frequencies are f Z 1 = 6.08 GHz and f Z 2 = 7.51 GHz, respectively, with coupling strength to the qubits of 17 MHz. 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 Refs. 7,22 , with the addition of the two zeroing registers. Figure 1B shows the characterization of the device by means of swap spectroscopy 7 . The computational capability of our architecture is displayed in Fig. 2A , where a 7-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, √ iSWAP, and iSWAP operations (step I, a to c) 22 . The corresponding density matrixρ (I) [ Fig ρ (III) [Fig. 2C (III)] is created between the qubits, using a sequence similar to the first operation (step III, a to c). In order to re-use 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 matrixρ (IV) of the zeroed two-qubit system is shown in Fig. 2C (IV). Once zeroed, the qubits can be used to read the memories (step V), allowing us to verify that, at the end of the algorithm, the stored state is still entangled. This is clearly demonstrated by the density matrix shown in 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 (IV). For the qubits, the storage time τ st is defined as the wait time at the end of step (I), prior to 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, while for the states stored in the memories it remains above 0.4 up to 1.5 µs. Most importantly, 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 taking 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 universal gates are a vital resource for the operation of the quCPU 2,3 . A variety of such gates have been implemented in superconducting circuits 5,[15] [16] [17] 19, 20 , with some recent demonstrations of quantum algorithms 5, 16 . 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 5, 16, 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 4 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 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 δ Q 1 B , which we term a "semi-resonant condition." In this process Q 1 acquires the phase whereg Q 1 B is the coupling frequency between |e1 and |f0 . The final step is to move the excitation from B back into Q 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 δ Q 1 B → ∞). These phases are measured by performing two Ramsey experiments on Q 1 for each value of the detuning δ Q 1 B , one with B in the |0 state, and the other with B in the |1 state. The relative phase between the Ramsey fringes corresponds to the value of φ for the CZ-φ gate 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 0 φ π, move the state in B back into Q 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 to create 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 readsρ Figure 3D shows the fidelity and entanglement of formation (EOF) 6 of two-qubit states generated using 70 values of φ. Figure 3E shows three examples ofρ φ for φ = 0.28, φ = π/2, and φ = π, respectively. 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 |p = 3 x=0 e i 2π xp/4 |x /2, corresponding to a 4×4 unitary operator. This operator can be fully characterized by means of quantum process tomography 2,5 , which allows us to obtain the χ p m matrix 2,5 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 threequbit 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 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 1 /2 CZ-π, CZ-π, and 1 /2 CZ-π, in an "M-shape" 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 , while 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 non-trivial 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 ; (vii), completing the Ramsey sequence on Q 2 with a second π/2-pulse with variable rotation axis relative to the pulse 6 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 while all off-diagonal elements of the matrix are zero. Each element τ XOR k can be expressed as a complex exponential e i φ |lmn , with |lmn ∈ M 3 . The phase φ |lmn can be either 0, when τ Among the eight values of φ |lmn , only seven are physically independent, as the element e i φ |gg0 can be factored, reducing the set of possible phases to φ |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 twelve 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 twelve phase differences, one can obtain the seven independent phases associated with the diagonal elements τ XOR k 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 1 /2 CZ-π gate between Q 1 and B shelves the |1 state from B to the non-computational state |f in Q 1 , where it remains until the second 1 /2 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 1 /2 CZ-π gates (cf. Fig. 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, where ε 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 superconducing 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.
Mariantoni et al. REF proposed the quantum von Neumann architecture including dedicated hardware for memories, zeroing registers, and a quantum bus.
Implementing the Quantum von Neumann Architecture with Superconducting Circuits
Science 334, 61-65 (2011)
Computer Science Physics Medicine