Supplementary Material for Implementing the Quantum von Neumann Architecture with Superconducting Circuits

Supplementary Material for
Implementing the Quantum von Neumann Architecture with Superconducting Circuits

Matteo Mariantoni, H. Wang,111Present address: Department of Physics, Zhejiang University, Hangzhou 310027, China.  T. Yamamoto, M. Neeley,222Present address: Lincoln Laboratory, Massachusetts Institute of Technology, 244 Wood Street, Lexington, MA 02420-9108, USA.
Radoslaw C. Bialczak, Y. Chen, M. Lenander, Erik Lucero, A. D. O’Connell,
D. Sank, M. Weides,333Present address: National Institute of Standards and Technology, Boulder, CO 80305, USA.  J. Wenner, Y. Yin, J. Zhao,
A. N. Korotkov, A. N. Cleland, John M. Martinis

Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA
Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan
Department of Electrical Engineering, University of California, Riverside, CA 92521, USA
California NanoSystems Institute, University of California,
Santa Barbara, California 93106-9530, USA

To whom correspondence should be addressed. E-mail: matmar@physics.ucsb.edu (M. M.);
martinis@physics.ucsb.edu (J. M. M.)

Implementing the Quantum von Neumann Architecture with Superconducting Circuits

Matteo Mariantoni, H. Wang, T. Yamamoto, M. Neeley, Radoslaw C. Bialczak, Y. Chen, M. Lenander, Erik Lucero, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, and John M. Martinis Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan Department of Electrical Engineering, University of California, Riverside, CA 92521, USA California NanoSystems Institute, University of California, Santa Barbara, CA 93106-9530, USA Present address: Department of Physics, Zhejiang University, Hangzhou 310027, China. Present address: Lincoln Laboratory, Massachusetts Institute of Technology, 244 Wood Street, Lexington, MA 02420-9108, USA. Present address: National Institute of Standards and Technology, Boulder, CO 80305, USA. To whom correspondence should be addressed. E-mail: matmar@physics.ucsb.edu (M. M.); martinis@physics.ucsb.edu (J. M. M.)

last updated: July 12, 2019
 

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 process fidelity, and the three-qubit Toffoli OR phase gate, with 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, 31, 3, 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 31, 3, the quantum Fourier transform 7, 9, and three-qubit Toffoli gates 12, 10. 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 15, 16, 17, 34, 19, 20 and the advanced control of both qubit and photonic quantum states 19, 20, 36, 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.

Figure 1: The quantum von Neumann architecture. (A) The quCPU (blue box) includes two qubits Q and Q and the bus resonator B. The quRAM (magenta boxes) comprises two memories M and M and two zeroing registers Z and Z. 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 36 for Q (left) and Q (right): Qubit excited state probability (color scale) vs. z-pulse amplitude (vertical axis) and delay time (horizontal axis), after exciting the qubit with a -pulse. At zero z-pulse amplitude the qubits are at their idle points, where they have an energy relaxation time  ns. A separate Ramsey experiment yields the qubits’ dephasing time  ns. By tuning the z-pulse amplitude, the qubit transition frequencies and can be varied between and  GHz. For z-pulse amplitudes indicated by B and M for Q, and by B and M for Q, the “chevron pattern” of a qubit-resonator interaction is observed 36. The transition frequencies of B, M, and M are  GHz,  GHz, and  GHz, respectively. From the chevron oscillation we obtain the qubit-resonator coupling strengths, which for both the resonator bus and the memories are  MHz (splitting) for the qubit transition, and faster for the transition (, , and are the three lowest qubit states) 22. For all resonators s. Swap spectroscopy also reveals that the qubits interact with several modes associated with spurious two-level systems. Two of them, Z and Z, are used as zeroing registers. Their transition frequencies are  GHz and  GHz, respectively, with coupling strength to the qubits of  MHz.

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 34, 19, 36, 22 Q and Q, connected through a coupling bus provided by a superconducting microwave resonator B. The quRAM comprises two superconducting resonators M and M that serve as quantum memories, as well as a pair of zeroing registers Z and Z, two-level systems that are used to dump quantum information. The chip geometry is similar to that in Refs. 36, 22, with the addition of the two zeroing registers. Figure 1B shows the characterization of the device by means of swap spectroscopy 36.

Figure 2: Programming the quantum von Neumann architecture. (A) Quantum algorithm comprising independent channels interacting through five computational steps. Dotted and solid lines represent channels in the ground and excited/superposition states, respectively. A black rectangle represents a -pulse; two crosses connected by a solid line a ; an open and a closed circle connected by a single arrow an iSWAP; oblique arrows indicate decay from a zeroing register. (B) Calibration of the zeroing gates. Each qubit is prepared in , interacts on resonance with its zeroing register for a time , and its probability measured, with plotted vs. (large and small blue circles). The solid green line is a decaying cosine fit to the data. The black arrows indicate the zeroing time for each qubit. (C) Density matrices of the Q-Q state for each step in A (scale key on bottom left). Grey arrows: Ideal state. Red and black arrows and black dots: Measured state (black arrows indicate errors). The off-diagonal elements of , , and have different angles because of dynamic phases 26. Fidelities: , , , , and . Concurrences: , , , , and . (D) Comparison of fidelity as a function of storage time for a Bell state stored in Q and Q (blue circles) vs. that stored in M and M (magenta squares; error bars smaller than symbols). The solid lines are exponential fits to data. (E) As in D, but for the concurrence . In D and E the vertical black dotted line indicates the time delay ( ns) associated with memory storage, with respect to storage in the qubits, due to the writing and reading operations (II) and (V) in A.

The computational capability of our architecture is displayed in Fig. 2A, where a -channel quantum circuit, yielding a dimensional Hilbert space, executes a prototypical algorithm. First, we create a Bell state between Q and Q using a series of -pulse, , and iSWAP operations (step I, a to c) 22. The corresponding density matrix [Fig. 2C (I)] is measured by quantum state tomography. The Bell state is then written into the quantum memories M and M by an iSWAP pulse (step II) 22, leaving the qubits in their ground state , with density matrix [Fig. 2C (II)]. While storing the first Bell state in M and M, a second Bell state with density matrix [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 and Q, for example to read out the quantum information stored in the memories M and M, the second Bell state has to be dumped 23. This is accomplished using two zeroing gates, by bringing Q on resonance with Z and Q with Z for a zeroing time , corresponding to a full iSWAP (step IV). Figure 2B shows the corresponding dynamics, where each qubit, initially in the excited state , is measured in the ground state after  ns. The density matrix 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 Fig. 2C (V).

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 35 of the Bell states stored in M and M are compared to those for the same states stored in Q and Q. The experiment is performed as in Fig. 2A, but eliminating steps (III) and (IV). For the qubits, the storage time 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 after  ns, while for the states stored in the memories it remains above up to s. Most importantly, after only  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 s (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.

Figure 3: The quantum Fourier transform. (A) (Left) Quantum logic circuit of a CZ- gate (enclosed in a grey box) for . The state of Q is indicated by a dashed line. The process where Q acquires the phase is represented by a pair of open/closed circles, connected by a single arrow in an arc shape. All other symbols are as in Fig. 2A. (Right) Shorthand symbol for the CZ- gate. Although the gate unitary matrix is symmetric, the symbol shows the asymmetric implementation of the gate. (B) Time-domain swaps between the states and , where we plot the probability (color scale) vs. interaction time and detuning . The solid black line indicates combinations of interaction time and detuning that completely depopulate the non-computational state. The three black dots on this line correspond to a CZ-, CZ-, and CZ- gate (see far right). The fourth black dot (outside the line) corresponds to a  CZ- gate (see bottom-left), where the state has been shelved to the non-computational state. (C) Phase acquired by Q as a function of . The blue dots indicate experimental data and the solid green line the theory of Eq. 1 26. (D) Fidelity (blue “+” symbols) and EOF (magenta “” symbols) of measured density matrices vs. . (E) (Left to Right) Density matrices , , and , obtained when , , and  rad in Eq. 2 (scale key on bottom left). The arrows are color-coded as in Fig. 2C. The measured fidelities are , , and , and EOF are (lower bound ), , and . (F) (Top-Left) Logic circuit for a two-qubit quantum Fourier transform and, (Bottom), real part of the corresponding matrix 31, 34. The process fidelity for the real and imaginary (not shown) part of is . The confidence intervals are estimated from measurements for , for and , and for .

Two-qubit universal gates are a vital resource for the operation of the quCPU 31, 3. A variety of such gates have been implemented in superconducting circuits 15, 16, 17, 34, 19, 20, with some recent demonstrations of quantum algorithms 16, 34. Control Z- (CZ-) 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 25, 16, 20, 34. However, CZ- gates are just a subset of the more general class of CZ- gates, obtained for the special case where the phase . In our architecture, the full class of CZ- gates, with from to , 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- gate (Left) and a shorthand symbol for the gate (Right). The logic circuit demonstrates the nontrivial case where qubits Q and Q are brought from their initial ground state to by applying a -pulse to each qubit. The excitation in Q is then transferred into bus resonator B, and Q’s transition brought close to resonance with B for the time required for a -rotation, where the states and are detuned by a frequency , which we term a “semi-resonant condition.” In this process Q acquires the phase 26

(1)

where is the coupling frequency between and . The final step is to move the excitation from B back into Q.

The time-domain swaps of between the states and are shown in Fig. 3B, where the solid black line indicates the detunings and corresponding interaction times used to generate any phase (ideally when ). These phases are measured by performing two Ramsey experiments on Q for each value of the detuning , one with B in the state, and the other with B in the state. The relative phase between the Ramsey fringes corresponds to the value of for the CZ- gate 26, as shown in Fig. 3C.

A more sophisticated version of this experiment is performed by initializing Q and Q each in the superposition state . We move Q’s state into B, perform a CZ- gate with , move the state in B back into Q, rotate Q’s resulting state by about the -axis, and perform a joint measurement of Q and Q. Ideally, this protocol permits to create two-qubit states ranging from a product state for to a maximally-entangled state for . In the two-qubit basis set , the general density matrix of such two-qubit states reads

(2)

Figure 3D shows the fidelity and entanglement of formation (EOF) 35 of two-qubit states generated using values of . Figure 3E shows three examples of for , , and , respectively.

The state generated using 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, followed by a CZ- gate between Q and Q, and finally a Hadamard on Q 31, 7, 9, as sketched in Fig. 3F (Top-Left). Representing the input state of the transform as (position) and the output as (momentum), assuming and the indexes and are integers, with , the output state , corresponding to a unitary operator. This operator can be fully characterized by means of quantum process tomography 31, 34, which allows us to obtain the matrix 31, 34 shown in Fig. 3F (Bottom).

Figure 4: Three-qubit gates: The XOR phase gate and the Toffoli-class M gate. (A) Quantum logic circuit for the XOR phase gate. (B) (Top) XOR-gate truth table. (Bottom) Ramsey fringes associated with the truth table, showing the probability of measuring Q in , vs. the Ramsey phase , for the control input states in . Black and magenta dots: phase. Blue and green dots: phase. The solid lines are least-squares fits to the data used to extract the truth-table phases. (C) Quantum phase tomography for the XOR gate: Phase , for each state . Black open boxes: Ideal values. Pink areas: Measured values with corresponding confidence intervals (black lines). (D) Quantum logic circuit for the M gate, implemented as a  CZ- gate (cf. Fig. 3B) between Q and B (half-dot/half-open circle connected by solid line), followed by a CZ- gate between Q and B, and a second  CZ- gate between Q and B. The dotted black lines connecting the two  CZ- gates indicate qubit shelving to the state. (E) As in panel B, but for the M gate. (F) As in C, but for the M gate.

Finally, by combining the CZ- and zeroing gates, we can implement a Toffoli-class gate 27, 12, 10, 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 AND Q, are in state . In our case, the control gate is applied conditionally if the controls Q OR Q are in . 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 and Q and the target is the bus resonator B, effectively acting as the third qubit (as only the states and of B are used). The XOR gate is realized as a series of two CZ- gates between the controls and the target, and the OR gate as the series  CZ-, CZ-, and  CZ-, in an “M-shape” configuration.

The truth table for the XOR gate is displayed in Fig. 4B (Top). The control qubits Q and Q are assumed to be in one of the states in , while the target B is in . The target acquires a phase , corresponding to a “true” result, only when the controls are in the state or . For the other non-trivial case , the target acquires phase, corresponding to a “false” result. This is due to the action of the two CZ- gates, giving a global phase when either of the controls is in , and a phase (equivalent to a phase) when both are in .

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 in the superposition state by means of a -pulse; (ii), moving the state from Q into B, thus creating a state in B; (iii), preparing Q and Q in each possible pair of control states in by means of -pulses; (iv), performing the XOR gate; (v), zeroing Q into Z at the end of the XOR gate; (vi), moving the final target state from B into the zeroed Q; (vii), completing the Ramsey sequence on Q with a second -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 and are inverted relative to the reference state , as expected from the XOR gate truth table.

In general, given the Q-Q-B basis set , the vector of the diagonal elements associated with the ideal unitary matrix of the XOR gate reads

(3)

while all off-diagonal elements of the matrix are zero. Each element can be expressed as a complex exponential , with . The phase can be either , when , or , when . Among the eight values of , only seven are physically independent, as the element can be factored, reducing the set of possible phases to , with .

In analogy to the truth-table for the target B, a table with four phase differences can also be obtained for the controls Q and Q, 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  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 acquired by the target B when the controls Q and Q are loaded in state . In this case, the action of the first  CZ- gate between Q and B shelves the state from B to the non-computational state in Q, where it remains until the second  CZ- gate. Moving the state of Q outside the computational space during the intermediate CZ- gate between Q and B effectively turns off the CZ- gate 29, 12. The target B thus only acquires a total phase due to the combined action of the two  CZ- gates (cf. Fig. 4D). The experimental truth table obtained from Ramsey fringes is shown in Fig. 4E (Bottom).

The vector of the diagonal elements associated with the ideal unitary matrix of the M gate is . 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,

(4)

where is the gate root-mean-square phase error, with an upper bound of . For the XOR gate we find that , and for the M gate .

Our results provide optimism for the near-term implementation of a larger-scale quantum processor 31, 3, 1 based on superconducing circuits. Our architecture shows that proof-of-concept factorization algorithms 31, 3, 5 and simple quantum error correction codes 31, 3, 6, 8 might be achievable using this approach.

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Acknowledgements

This work was supported by IARPA under ARO award W911NF-08-1-0336 and under ARO award W911NF-09-1-0375. M. M. acknowledges support from an Elings Postdoctoral Fellowship. Devices were made at the UC 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.

Author Contributions

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 co-wrote the manuscript.

This PDF file includes:

Materials and Methods

Figs. S1 to S12

Tables S1 to S3

References

Materials and Methods

Statistical errors

In this section, we analyze the statistical properties of the experimental data shown in the main text. First, we explain how to simulate statistical errors. This procedure was used to estimate the confidence intervals for the data of Fig. 2 in the main text. Second, we describe how statistical errors were obtained from statistical ensembles of independent measurements. This procedure was used for the data of Fig. 3 in the main text. Third, we discuss the estimation of statistical errors due to fits to the data. This procedure was used for the data of Fig. 4 in the main text.

Simulation of statistical errors

In this subsection, we discuss two important sources of statistical errors in our data: Errors associated with qubit’s measurement (binomial-type errors) and errors due to jitter/fluctuations in the electronics (phase errors). Assuming binomial-type and phase errors, we describe the procedures used to simulate the confidence intervals for the elements and metrics of the density matrices shown in Fig. 2C of the main text.

Figure S1: Analysis of phase errors. (A) Phase angle associated with the off-diagonal elements of the matrix of Eq. S7 plotted vs. time. The time axis indicates when the QST of each density matrix was completed. The dashed black lines are a guide-to-the-eye showing an increase with time in the data scatter. (B) Histogram associated with the time-trace data in A, plotting the number of elements in the time-trace vs. the phase angle . The solid green line is a fit to a normal distribution with mean value of  rad and standard deviation  rad. The window is indicated. (C) Time-bin average of the data in A, showing the value of for each time-bin of  min, for a total of bins (blue diamonds). The bins are indicated by vertical dotted grey lines, which extend to A for clarity. The solid magenta line is a linear fit to the data. This fit was used to estimate the phase errors associated with QST measurements (tomo and octomo).
  1. Binomial-type errors are inherent to our qubit measurement process, where the measurement is repeated a fixed number of times , each measurement trial has two possible outcomes, i.e., qubit being in the ground state with probability or in the excited state with probability , the probability is to good approximation the same for each trial, and the trials can be considered to be statistically independent. The measurement outcome associated with is counted as , and that associated with as . Under these assumptions, the qubit measurement process can be described by a binomial distribution.

    Given a statistical sample consisting of measurement outcomes (i.e., a statistical sample from a Bernoulli distribution with parameter ), the maximum likelihood estimator of (i.e., the estimated probability) is given by

    (S5)

    where represents the -th outcome among the measured. There are several ways to compute a confidence interval for the parameter . The most common result is based on the approximation of the binomial distribution with a normal distribution. This represents a good approximation in our experiments, where the number of measurements is large (typically ). In this case, it can be shown that a confidence interval for the parameter is given by

    (S6)

    where is the percentile of a standard normal distribution. For example, for a (%) confidence interval, we set , so that . When analyzing our data we approximate the percentile with , thus obtaining a slightly wider confidence interval;

  2. Phase errors are mostly due to the phase jitter/fluctuations in the room-temperature cables and electronics used to measure the qubits. In order to quantify such errors, the following experiment was performed. First, we initialized one of the two qubits, e.g., qubit Q, in the ground state, ; second, we applied to Q a unitary rotation about the -axis, , bringing the qubit into the state . This state is characterized by the density matrix

    (S7)

    which represents a “phase-sensitive” state due to the presence of nonzero off-diagonal elements, thus allowing us to measure the phase properties of our setup. In fact, if the setup (cables and electronics) were ideal, the phase associated with the off-diagonal elements of the matrix of Eq. S7 would be zero. We can thus assume that any deviation from a zero phase corresponds to a phase error; third, we performed a single-qubit quantum state tomography (QST) on Q, making possible to measure experimentally . Using our typical settings for a single-qubit QST 30, the time needed for each QST was approximately  s; fourth, we repeated a QST measurement every  s for a total time of  minutes, corresponding to measured density matrices; finally, we plotted as a function of time. The so-obtained time trace is shown in Fig. S1A. Besides negligible slow-varying oscillations in the time trace [independent tests have shown that these oscillations might be due to temperature changes in the room-temperature cables (data not shown)], the overall histogram associated with the trace is approximately normally distributed about a mean value of  rad, with standard deviation  rad (cf. Fig. S1B). However, we notice a general increase in the scatter of the time-trace data, as indicated by the dashed black lines in Fig. S1A. We thus divide the time trace in sub-traces (time bins) with a time length of  min each, compute the standard deviation for each sub-trace, and plot the so-obtained standard deviations as a function of time. The result is displayed in Fig. S1C, where the data is overlayed with a linear fit.

The plot of Fig. S1C is useful in determining the phase errors associated with different types of two-qubit QST, as well as quantum process tomography (QPT) 31, 32, 33, 34. In fact, two-qubit QST can be realized either by applying to each qubit the set of three unitary operations ( is the identity matrix, a unitary rotation about the -axis, and a unitary rotation about the -axis), which we call “tomo,” or the set of six unitary operations ( is a unitary rotation about the -axis, a unitary rotation about the -axis, and a unitary rotation about the -axis), which we call “octomo.”

In the case of two-qubit tomo, the number of operations that must be applied to the pair of qubits is given by the permutations of the allowed set of unitary operations, . This number multiplied by the possible joint probabilities for a two-qubit system, , and (where, e.g., is the probability to measure the first qubit in the ground state with the second qubit in the excited state) gives a total of probabilities. In the case of octomo, the total number of probabilities is given by the permutations of unitary operations for qubits, , times the possible joint probabilities for a two-qubit system, for a total of probabilities.

In the experiments, the maximum likelihood estimator for each of the four probabilities , and is obtained from the outcome of measurements. We note that, in a joint two-qubit measurement each outcome consists of numbers obtained simultaneously, where each number can be either or . The statistical sample consisting of two-qubit joint measurements will be hereafter defined as , with . Similarly to Eq. S5, the maximum likelihood estimator (i.e., the estimated probability) for each of the four probabilities , and can thus be obtained from

(S8)

where represents the -th outcome among the measured.

For a given , the four possible , i.e., , , , and , are measured simultaneously (with ). Hence, the effective number of events that has to be measured for each tomo is , and for each octomo .

We typically measure events per second, and repeat each measurement times. As a consequence, a two-qubit tomo takes approximately  min, and a two-qubit octomo approximately  min.

All data displayed in Fig. 2C of the main text were obtained using tomo, while all data in Fig. 3, D and E, were obtained using octomo. All density matrices used to reconstruct the matrix of Fig. 3F in the main text were also obtained with octomo. The standard deviation due to phase errors can be estimated in each case by looking up the fit in Fig. S1C.

Considering for example a two-qubit octomo with , the statistical properties of the resulting density matrix and of the corresponding metrics [fidelity , negativity , concurrence , and entanglement of formation ; cf. Ref. 35 and references therein for an extensive description of these metrics] are obtained as follows:

  1. The probabilities associated with two-qubit octomo are estimated according to Eq. S8. As explained above, this corresponds to a total of estimated probabilities. To simplify the notation, we will hereafter refer to these probabilities as , with ;

  2. The estimated probabilities are corrected for measurement errors [cf. Refs. 36 and 37 for our standard procedures to correct for measurement errors in the case of one and two qubits, respectively]. The corrected probabilities are stored as a column vector;

  3. For each probability , the binomial standard deviation defined in Eq. S6 is calculated, thus obtaining, in the case of octomo, different standard deviations;

    Figure S2: Confidence intervals for a density matrix and its metrics. (A) Histogram associated with the -th probability of the vector of probabilities , plotting the number of elements among the probabilities obtained in point () vs. the corresponding value of the probability . The data refers to the octomo for the state of Fig. 3E in the main text. The solid green line is a fit to a normal distribution. (B) Histogram for the real part of the matrix element with mean value for the state of Fig. 3E in the main text. The solid green line is a fit to a normal distribution. The window is indicated, where is one standard deviation.
  4. For each of the standard deviations calculated in (), a set of random numbers picked from a normal distribution with zero mean value and standard deviation is generated. This results in a matrix of random numbers. Typically, .

    By summing each column of such a matrix to the column vector containing the estimated probabilities , we obtain a matrix of probabilities, where each column simulates the result of a different QST experiment.

    For example, Fig. S2A shows the histogram associated with the -th probability of the vector of probabilities in the case of the octomo for the state of Fig. 3E in the main text;

  5. Each column of the matrix of probabilities obtained in point () is inverted by following the usual QST rules 30, 37. This allows us to find the corresponding density matrix , with , thus obtaining density matrices associated with one state;

  6. Physicality constraints are enforced on each, generally unphysical, density matrix by means of the MATLAB packages SeDuMi  and YALMIP (semidefinite programming) 38. The physical constraints are such that each final - physical - density matrix should have unit trace and be positive semidefinite.

    In order to obtain the mean physical density matrix associated with the physical density matrices and the corresponding standard deviations, we calculate the mean value and standard deviation of the real and imaginary part of each matrix element for the matrices . The mean physical matrix will thus have elements (with ), each of them (real and imaginary part) characterized by a given standard deviation. Figure S2B shows the histogram for the real part of the matrix element with mean value for the state of Fig. 3E in the main text. As expected, the distribution is approximately Gaussian with a confidence interval .

    The knowledge of the matrices also allows us to estimate the confidence intervals for the relevant metrics characterizing the state : , , , and . This can easily be accomplished by calculating the metrics for each , thus obtaining values for each metric, and then computing the mean value and standard deviation of the values associated with each metric.

We can follow a similar procedure to account for phase errors. We now pick two independent sets of random numbers from a normal distribution with zero mean value and standard deviation (with opportunely estimated from Fig. S1C depending on whether a tomo or octomo was used), thus generating two sets of phase errors, and , with . In order to simulate phase errors acting independently on each qubit, we apply the unitary rotation

(S9)

to a measured density matrix , thus obtaining the -th unphysical density matrix

(S10)

We can then proceed as in step (6) above and obtain a mean physical density matrix and its statistical properties, as in the case of binomial-type errors. This allows us also to find the metrics associated with and their statistical properties. Notice that the unitary transformation of Eq. S9 simulates random rotations along the -axis of both qubit and qubit .

The total mean physical density matrix is finally obtained by averaging the mean physical density matrix obtained in the case of binomial-type errors and the matrix obtained in the case of phase errors. The same applies to the mean values of all metrics. The corresponding standard deviations are found by summing in quadrature the values obtained in the case of binomial-type and phase errors. For example, the numerical value with confidence interval of each element of the density matrices in Fig. 2C of the main text were obtained following this procedure. These numbers are reported in Table S1.

Incidentally, we found that phase errors do not add any significant contribution to the confidence intervals of the density matrix elements and of their metrics.

Notice that, the reason why we decided to simulate the statistical properties of the data in Fig. 2 of the main text is because we only had independent measurements of these data. Such a statistical ensemble is obviously insufficient to obtain reliable confidence intervals, which, thus, needed to be simulated.

Experimental estimation of statistical errors

In the case of the density matrices in Fig. 3E and of the matrix of the quantum Fourier transform in Fig. 3F of the main text we had ensembles of independent measurements large enough to allow the confidence intervals estimation directly from the data.

In particular, the density matrix in the left panel of Fig. 3E is the average of a statistical ensemble of independent measurements, and the density matrices and in the center and right panels of Fig. 3E, respectively, are the average of an ensemble of independent measurements. The standard deviation of each matrix element (real and imaginary part) as well as the mean value and standard deviation of all metrics can easily be estimated from such statistical ensembles.

Finally, the matrix of Fig. 3F is the average of an ensemble of independent measurements. This allows us to estimate the mean value and standard deviation of the process fidelity associated with the quantum Fourier transform (cf. main text).

Statistical errors of fitted parameters

The confidence intervals associated with the quantum phase tomography data shown in Fig. 4, C and F, of the main text are dominated by the statistical errors of the coefficients fitted from the data in Fig. 4, B and E, of the main text. In particular, the coefficient of interest is the phase of each curve in Fig. 4, B and E (or, more in general, of each curve in Fig. S12, C and D).

We remind that the error vector associated with the vector of coefficients fitted to a curve is given by the square root of the vector of the diagonal elements from the estimated covariance matrix of the coefficient estimates, . Here, is the Jacobian of the fitted values with respect to the coefficients, is the transpose of , and is the mean squared error. This procedure allows us to estimate the errors associated with the fitted phases. These errors propagate through the quantum tomography process (cf. section on “Quantum phase tomography” at the end of these Methods), finally turning into the confidence intervals reported in Fig. 4, C and F, of the main text.

Definition of the qubit reference frame

In this section, we briefly explain the concepts of reference frame and reference clock rate associated with a qubit. These concepts will be useful in understanding the dynamic phases acquired by the qubits when programming the quantum von Neumann architecture as well as the sequences used to tune up the CZ- gates and the XOR and M gate.

In the two-level approximation 39, the Hamiltonian of a phase qubit can be written as

(S11)

with ground state and excited state , and eigenenergies and , respectively. In Eq. S11, represents the qubit transition frequency, which can be tuned by means of z-pulses with amplitude , and is the usual spin Pauli operator. At the beginning of a CZ- gate, each qubit is initialized in at the so-called idle point, which corresponds to a z-pulse amplitude . The qubit transition frequency at the idle point is thus given by .

In order to prepare a qubit in the excited state or in a linear superposition , the qubit has to be driven by a microwave pulse. The Hamiltonian governing the interaction between the qubit and the microwave driving is given by 40

(S12)

where is the time-dependent driving amplitude expressed in unit hertz, the driving frequency, the usual spin Pauli operator, the time, and an arbitrary phase delay. By calibrating the microwave pulse such that the phase delay , we can rewrite the driving Hamiltonian as

(S13)

By combining the qubit Hamiltonian of Eq. S11 and the driving Hamiltonian of Eq. S13, we obtain the total Hamiltonian of the driven system, .

In our experiments the driving frequency is a fixed parameter that is set to be equal to the qubit transition frequency at the idle point 41,

For a given qubit, the microwave driving represents the reference frame associated with that qubit, with reference clock rate given by . Defining the detuning between the z-dependent qubit transition frequency and the reference clock rate as , the qubit-driving Hamiltonian can be expressed in the uniformly rotating reference frame by applying the unitary rotation  42. The rotated Hamiltonian is thus given by

(S14)

where the counter-rotating terms have been already neglected. The dynamics associated with the pulse sequences used to tune up the CZ- gates and the XOR and M gate can be understood by following the time-evolution of the Hamiltonian of Eq. S14. In particular, the Hamiltonian describes the dynamic phases acquired by the qubits when they are brought outside their reference frame (i.e., qubit rotations about the -axis). As it will appear clear when describing the tune-up sequences of the CZ- gates and of the XOR and M gate, in the experiments we always compensate for such dynamic phases.

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