Low depth measurement-based quantum computation with qudits ($d$-level systems) is investigated and a precise relationship between this powerful model and qudit quantum circuits is derived in terms of computational depth and size complexity. To facilitate this investigation a qudit `unbounded fan-out' circuit model, in which a qudit may be quantum-copied into an arbitrary number of ancillas in a single time-step, is introduced and shown to be capable of implementing interesting $n$-qudit unitaries in constant depth. A procedure for reducing the quantum computational depth in the measurement-based model is then proposed and using this it is then shown that there is a logarithmic depth separation between the depth complexity of qudit measurement-based computation and circuits composed of gates act on a bounded number of qudits. The relationship is made precise by showing that the depth complexity of the qudit measurement-based model is exactly equivalent to that of unbounded fan-out circuits. These results illustrate that the well-known advantages inherent in qubit measurement-based quantum computation are also applicable to the higher-dimensional generalisation. As qudits are both naturally available and have been shown to provide fundamental advantages over binary logic encodings, this then suggests that the qudit measurement-based model is a particularly appealing paradigm for universal quantum computation.
Low depth measurement-based quantum computation with qudits (d-level systems) is investigated and a precise relationship between this powerful model and qudit quantum circuits is derived in terms of computational depth and size complexity. To facilitate this investigation a qudit `unbounded fan-out' circuit model, in which a qudit may be quantum-copied into an arbitrary number of ancillas in a single time-step, is introduced and shown to be capable of implementing interesting n-qudit unitaries in constant depth. A procedure for reducing the quantum computational depth in the measurement-based model is then proposed and using this it is then shown that there is a logarithmic depth separation between the depth complexity of qudit measurement-based computation and circuits composed of gates act on a bounded number of qudits. The relationship is made precise by showing that the depth complexity of the qudit measurement-based model is exactly equivalent to that of unbounded fan-out circuits. These results illustrate that the well-known advantages inherent in qubit measurement-based quantum computation are also applicable to the higher-dimensional generalisation. As qudits are both naturally available and have been shown to provide fundamental advantages over binary logic encodings, this then suggests that the qudit measurement-based model is a particularly appealing paradigm for universal quantum computation.
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.
The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by quantum shallow circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice QNC0/qpoly , and AC0[p] , which is the class of classical constant-depth circuits with unbounded-fan in and MODp gates. Additionally, we show a separation between QAC0 , the circuit class containing Toffoli gates with unbounded control, and AC0[p] , when QAC0 is augmented with additional mid-circuit measurements and classical fanout. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fanout gates, while relying solely on finite quantum gate sets. Equivalently, this yields a separation between AC0[p] and [QNC0,AC0]2 , i.e., shallow quantum circuits interleaved with simple classical computation. Secondly, we consider QNC0 circuits with infinite-size gate sets. We show that these circuits, along with quantum prime modular gates or classical prime modular gates in combination with classical fanout, can implement threshold gates, showing that QNC0[p]=QTC0 . Finally, we also show that in the infinite-size gate set case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.
It is well-known that every n-qubit unitary can be implemented by a 2O(n)-depth quantum circuit using single- and two-qubit gates. It has been open whether exponential depth is *necessary* for general unitaries, even when allowing an unlimited number of ancilla qubits. Here we show, perhaps surprisingly, that all unitaries can be implemented exactly by a circuit of one- and two-qubit gates of depth poly(n) with 2O(n) ancilla qubits. In other words, every n-qubit unitary can be parallelized to polynomial depth. In fact, our depth bound is *linear* in n, which is the best possible, and an exponential improvement on the previous best bound of 2n/2 due to Rosenthal [TQC 2022, Quantum 2026]. Moreover, if we allow unbounded fan-out gates, these circuits can be further reduced to *constant* depth. Our construction takes advantage of a novel relationship connecting the unitary synthesis problem of Aaronson and Kuperberg to locally-decodable codes and private information retrieval from complexity theory and cryptography, and has a natural interpretation in bosonic quantum computation.
A model of universal quantum computation with qudits (d-level systems) is presented in which the gates are implemented on a register entirely via fixed-form sequential interactions with ancillary qudits followed by adaptive, variable basis measurements of these ancillas. The model maintains determinism via classical feed-forward of measurement outcomes in a similar fashion to the qudit one-way model. An embedding of the qudit one-way model is given which demonstrates that the hybrid quantum-classical advantages of one-way quantum computation are also inherent in the model presented here, including the ability to implement any Clifford gate in constant depth using multiple ancillas in parallel. A model which requires only fixed basis measurements is then presented, but due to the lack of adaptive controls it can only implement universal quantum computation in a stochastic sense. Interestingly, this scheme may then be altered to recover determinism by replacing ancilla measurements with ancilla preparation. These models provide an intuitive setting for understanding the interplay between the benefits gained from classical controls, measurements and state preparation in the context of multi-valued logic ancilla-based quantum computation.
We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant δ>0, it ε-approximates an arbitrary single-qubit gate using O(log1+δ(1/ε)) clean ancillae, Hadamard and T single-qubit gates, O(log(1/ε))-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard, T, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth O(loglog(1/ε)), and we prove a matching Ω(loglog(1/ε))-depth lower bound. Overall, we establish that Θ(loglog(1/ε))-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis.
These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-d QAC circuit can be simulated in depth O(d) using only Hadamard, X, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity∈/QAC0 to proving a Parity lower bound against circuits consisting only of Hadamard, X, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.
This paper shows that, in measurement-based quantum computation, it is possible to write any quantum oracle implementing a classical function in constant depth. The result is shown through the equivalence between MBQC and the circuit model where arbitrary rotations along Z axis and unbounded fan-outs are elementary operations. A corollary of this result is that disjunction can be implemented exactly in constant-depth, answering an open question of Høyer and Špalek.