Experimental accreditation of outputs of noisy quantum computers

We provide and experimentally demonstrate an accreditation protocol that upper bounds the variation distance between noisy and noiseless probability distributions of the outputs of arbitrary quantum computations. We accredit the outputs of 24 quantum circuits executed on programmable superconducting hardware, ranging from depth-9 circuits on 10 qubits to depth-21 circuits on 4 qubits. Our protocol requires implementing the “target” quantum circuit along with a number of random Clifford circuits and subsequently postprocessing the outputs of these Clifford circuits. Importantly, the number of Clifford circuits is chosen to obtain the bound with the desired confidence and accuracy and is independent of the size and nature of the target circuit. We thus demonstrate a practical and scalable method of ascertaining the correctness of the outputs of arbitrary-sized noisy quantum computers—the ultimate arbiter of the utility of the computer itself.

Figure 1

Experimental bounds on the VD [rhs of Eq. (1)] provided by our AP. (a) Accreditation of GHZ and QFT circuits. (b) Accreditation of six-qubit random circuits. In (a) the numbers inside the rectangles indicate the depths of the various target circuits. The bounds in (a) are calculated by implementing $v=450$ trap circuits; those in (b) are calculated by implementing $v=900$ trap circuits. The bars correspond to confidence levels above $95\%$ on our estimates of $2p_{\text{inc}}$; specifically, in (a) we set $\theta =13\%$ and $\alpha =95\%$, and in (b) we set $\theta =9\%$ and $\alpha =95\%$ [see Eq. (2)].

Figure 2

(a) Example of a target circuit. The target circuit must be compiled into $m$ cycles of one-qubit gates, each one (apart from the last one) followed by a cycle of $cZ$ gates (giving circuit depth $d=2m-1$). Input qubits are in state $|0\rangle$, and measurements are in the Pauli-$Z$ basis. (b) Example of a trap circuit for the target circuit in (a). The trap circuit is obtained by replacing the one-qubit gates in the target circuit with one-qubit Clifford gates. Neighboring cycles of one-qubit gates can be recompiled into a single cycle. Thus, the trap circuit has the same circuit depth as the target.

Figure 3

A graphical representation of the connectivity in (a) ibmq_johannesburg and (b) ibmq_paris. The circles represent qubits; the edges represent the available entangling gates.

Figure 4

(a) Our largest GHZ circuit. After adding the QOTP, this circuit contains nine cycles of noncommuting gates. (b) Our largest QFT circuit. We apply the circuit to state ${|+\rangle }^{\otimes 4}$. Each two-qubit gate is a controlled-phase gate, which we decompose into two $cZ$ gates interleaved by one-qubit gates [33]. After adding the QOTP, this circuit contains 21 cycles of noncommuting gates. (c) Our six-qubit depth-nine pseudorandom circuit. The various gates $V_{i,j}$ are random one-qubit gates.

Figure 5

Values of VD (red points) and $p_{\text{inc}}$ (blue points) measured in the experiment where the target circuit generates a six-qubit GHZ state. Each job contains 900 circuits (450 target circuits and 450 independently chosen trap circuits). The bars correspond to confidence levels above $95\%$; specifically, we set $\theta /2=6.5\%$ and $\alpha \gtrsim 95\%$ [see Eq. (2)].

Figure 6

Examples of trap circuits affected by errors [the faulty gates and measurements are highlighted in red (gray)]. (a) Single-cycle pattern affecting a single qubit. Patterns of this type are detected with a probability of $50\%$ (see Ref. [24]). (b) Multicycle pattern. Patterns of this type are detected with a probability greater than $50\%$.

Figure 7

The values of $P_{\text{trap}}(H_{\overline{s}}=h)$ calculated using the bit-flip model defined in Eq. (4) (striped bars) and those measured in the experiments with pseudorandom circuits (solid bars); for $h\ge 3$ the $y$ axis is rescaled. At any given depth, the values of $P_{\text{trap}}(H_{\overline{s}}=h)$ are measured using the outputs of 17 980 traps. In (a) we set $p_{\text{flip}}=2.3\%$, which coincides with the error rate provided for the measurements by the back end. In (b) we set $p_{\text{flip}}=7.6\%$.

Figure 8

The best value provided by the original AP [Eq. (A37)] and the bound provided by the refined AP [rhs of Eq. (1)] as functions of $p_{\mathrm{inc}}$.