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The ability to perform measurements in the middle of a quantum circuit is a powerful resource. It underlies a wide range of applications, from remote state preparation to quantum error correction. Here we apply mid-circuit measurements for a particular task: demonstrating quantum computational advantage. The goal of such a demonstration is for a quantum device to perform a computational task that is infeasible for a classical device with comparable resources. In contrast to existing demonstrations, the distinguishing feature of our approach is that the classical verification process is efficient, both in asymptotic complexity and in practice. Furthermore, the classical hardness of performing the task is based upon well-established cryptographic assumptions. Protocols with these features are known as cryptographic proofs of quantumness. Using a trapped-ion quantum computer, we perform mid-circuit measurements by spatially isolating portions of the ion chain via shuttling. This enables us to implement two interactive cryptographic proofs of quantumness, which when suitably scaled to larger systems, promise the efficient verification of quantum computational advantage. Our methods can be applied to a range of interactive quantum protocols.

Many-body open quantum systems balance internal dynamics against decoherence and measurements induced by interactions with an environment1,2. Quantum circuits composed of random unitary gates with interspersed projective measurements represent a minimal model to study the balance between unitary dynamics and measurement processes3–5. As the measurement rate is varied, a purification phase transition is predicted to emerge at a critical point akin to a fault-tolerant threshold6. Here we explore this purification transition with random quantum circuits implemented on a trapped-ion quantum computer. We probe the pure phase, where the system is rapidly projected to a pure state conditioned on the measurement outcomes, and the mixed or coding phase, where the initial state becomes partially encoded into a quantum error correcting codespace that keeps the memory of initial conditions for long times6,7. We find experimental evidence of the two phases and show numerically that, with modest system scaling, critical properties of the transition emerge.

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system1,2. These extra degrees of freedom enable the detection and correction of errors, but also increase the control complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while controlling the logical qubit, and are essential for realizing error suppression in practice3–6. Although fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. Here we experimentally demonstrate fault-tolerant circuits for the preparation, measurement, rotation and stabilizer measurement of a Bacon–Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault-tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6 per cent and a Clifford gate error of 0.3 per cent after offline error correction. In addition, we prepare magic states with fidelities that exceed the distillation threshold7, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant control. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.