Constant-depth fan-out with real-time feedforward on a superconducting quantum processor
Yongxin Song, Liberto Beltrán, Ilya S. Besedin, Michael Kerschbaum, Marek Pechal, François Swiadek, Christoph Hellings, Dante Colao Zanuz, Alexander Flasby, Jean-Claude Besse, Andreas J. Wallraff
When using unitary gate sequences, the growth in depth of many quantum circuits with output size poses significant obstacles to practical quantum computation. The quantum fan-out operation, which reduces the circuit depth of quantum algorithms such as the quantum Fourier transform and Shor’s algorithm, is an example that can be realized in constant depth independent of the output size. Here, we demonstrate a quantum fan-out gate with real-time feedforward on up to four output qubits using a superconducting quantum processor. By performing quantum state tomography on the output states, we benchmark our gate with input states spanning the entire Bloch sphere. We decompose the output-state error into a set of independently characterized error contributions. We extrapolate our constant-depth circuit to offer a scaling advantage compared to the unitary fan-out sequence beyond 25 output qubits with feedforward control, or beyond 17 output qubits if the classical feedforward latency is negligible. Our work highlights the potential of midcircuit measurements combined with real-time conditional operations to improve the efficiency of complex quantum algorithms.
When using unitary gate sequences, the growth in depth of many quantum circuits with output size poses significant obstacles to practical quantum computation. The quantum fan-out operation, which reduces the circuit depth of quantum algorithms such as the quantum Fourier transform and Shor’s algorithm, is an example that can be realized in constant depth independent of the output size. Here, we demonstrate a quantum fan-out gate with real-time feedforward on up to four output qubits using a superconducting quantum processor. By performing quantum state tomography on the output states, we benchmark our gate with input states spanning the entire Bloch sphere. We decompose the output-state error into a set of independently characterized error contributions. We extrapolate our constant-depth circuit to offer a scaling advantage compared to the unitary fan-out sequence beyond 25 output qubits with feedforward control, or beyond 17 output qubits if the classical feedforward latency is negligible. Our work highlights the potential of midcircuit measurements combined with real-time conditional operations to improve the efficiency of complex quantum algorithms.
When using unitary gate sequences, the growth in depth of many quantum circuits with output size poses significant obstacles to practical quantum computation. The quantum fan-out operation, which reduces the circuit depth of quantum algorithms such as the quantum Fourier transform and Shor's algorithm, is an example that can be realized in constant depth independent of the output size. Here, we demonstrate a quantum fan-out gate with real-time feedforward on up to four output qubits using a superconducting quantum processor. By performing quantum state tomography on the output states, we benchmark our gate with input states spanning the entire Bloch sphere. We decompose the output-state error into a set of independently characterized error contributions. We extrapolate our constant-depth circuit to offer a scaling advantage compared to the unitary fan-out sequence beyond 25 output qubits with feedforward control, or beyond 17 output qubits if the classical feedforward latency is negligible. Our work highlights the potential of midcircuit measurements combined with real-time conditional operations to improve the efficiency of complex quantum algorithms.
Dynamic quantum circuits combine mid-circuit measurement with classical feed-forward, enabling circuit constructions with reduced entangling-gate depth. Here, we investigate their use in Quantum Imaginary Time Evolution (QITE), where circuit depth and parameter growth limit practical implementations of ground-state preparation. For dense classical optimization Hamiltonians, we introduce a reduced-parameter QITE ansatz that restricts entanglement generation via a small set of control qubits, enabling each QITE layer to be implemented with constant two-qubit gate depth using fan-out-based dynamic circuits. In noiseless simulations of exact cover and set partitioning instances, the reduced ansatz yields a higher success probability than standard QITE approaches. We implement unitary, dynamic fan-out, and semi-classical adaptive variants on IBM superconducting hardware. The semi-classical variant performs favorably to the unitary implementation, while the fully dynamic construction exposes the trade-offs between entangling-depth reduction and measurement and feed-forward overhead associated to dynamic circuit implementations. Using a fidelity threshold of 0.5 relative to the noiseless QITE ansatz, we show that dynamic fan-out based QITE would outperform unitary implementations on current devices when the measurement and two-qubit gate errors are reduced by 65% and the feedback latency is halved.
The depth of quantum circuits is a critical factor when running them on state-of-the-art quantum devices owing to their limited coherence times. Reducing circuit depth decreases noise in near-term quantum computations and reduces overall computation time. This also benefits fault-tolerant quantum computations. Here, we show how to reduce the depth of quantum subroutines that typically scale linearly with the number of qubits, such as quantum fan-out and long-range CNOT gates, to a constant depth using mid-circuit measurements and feedforward operations, while only requiring a 1D line topology. We compare our protocols with existing ones to highlight their advantages. Additionally, we verify the feasibility by implementing the measurement-based quantum fan-out gate and long-range CNOT gate on real quantum hardware, demonstrating significant improvements over their unitary implementations.
Mid-circuit measurements with feedforward reduce the optimal worst-case quantum depth for exact n-qubit unitary synthesis from Θ(n) to O(1), using arbitrary one- and two-qubit gates and unrestricted ancillary space. Measurements and resets without feedforward remain subject to the circuit causal-cone bound. Our deterministic construction uses single-qubit measurements, O(1) rounds of parity feedforward, and O(n4nlog(n+2)) total qubits, with all ancillas starting and ending in ∣0⟩. Classical processing is not counted in quantum depth. We teleport only the matrix error of an approximation by Nehoran and Yuen. An exactly prepared ancillary state encodes this error, whose action is added back by coherent interference. At operator-norm error 2−n, one step of amplitude amplification suffices. Coherent fanout circuits achieve the same constant-depth bound, while circuits using only one- and two-qubit gates attain depth Θ(n) with O(4nlog2(n+2)) gates. Generic targets require Ω(4n/(h+1)) qubits at depth h, even under unrestricted classical control. Our depth-width trade-offs meet this bound within a factor O(n2) for 1≤h≤2n.
Classical circuits with unbounded fan-in can compute any Boolean function in constant depth when their size is unrestricted. We ask whether removing the restrictions on circuit size and ancillary qubits also allows quantum circuits built from arbitrary single-qubit gates and generalised Toffoli gates to implement every unitary in constant depth.
We give exact constant-depth constructions for arbitrary permutations of computational basis states, diagonal unitaries and the preparation of arbitrary pure states. These connect quantum state preparation to reversible classical computation and the preparation of probability distributions. With fanout in the gate set, they use exponentially many gates and ancillary qubits and return all ancillary qubits to zero. Replacing fanout by an exact circuit over the original gate set preserves constant depth, although the size bounds can become doubly exponential.
The implementation of arbitrary unitaries in constant depth remains open. We give equivalent formulations in terms of copying the vectors of a specified orthonormal basis, extracting their labels and implementing restricted families of unitaries. We also reduce arbitrary unitary implementation to that of traceless unitary involutions using one additional clean qubit. With adaptive measurements, gate teleportation gives depth proportional to the level of a gate in the Clifford hierarchy.
Towards arbitrary unitary implementation in constant depth, we use port-based teleportation: for input dimension d and M≥d2−1 ports, we construct a unitary circuit of depth O(d), independent of M and including resource preparation and port selection, with entanglement fidelity at least (1−(d2−1)/(2M))2. Thus, at fixed d, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on d can also be removed remains open.
We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNCf0) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fan-in gates. If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth.