Transformations of quantum channels, such as the transpose, complex conjugate, and adjoint, are fundamental to quantum information theory. Given access to an unknown channel, a central problem is whether these transformations can be implemented physically with quantum supermaps. While such supermaps are known for unitary operations, the situation for general quantum channels is fundamentally different. In this work, we establish a strict hierarchy of physical realizability for the transposition, complex conjugation, and adjoint transformation of an unknown quantum channel. We present a probabilistic protocol that exactly implements the transpose with a single query. In contrast, we prove no-go theorems showing that neither the complex conjugate nor the adjoint can be implemented by any completely positive supermap, even probabilistically. We then overcome this impossibility by designing a virtual protocol for the complex conjugate based on quasi-probability decomposition, and show its optimality in terms of the diamond norm. As a key application, we propose a protocol to estimate the expectation values resulting from the Petz recovery map of an unknown channel, achieving an improved query complexity compared to existing methods.
Transformations of quantum channels, such as the transpose, complex conjugate, and adjoint, are fundamental to quantum information theory. Given access to an unknown channel, a central problem is whether these transformations can be implemented physically with quantum supermaps. While such supermaps are known for unitary operations, the situation for general quantum channels is fundamentally different. In this work, we establish a strict hierarchy of physical realizability for the transposition, complex conjugation, and adjoint transformation of an unknown quantum channel. We present a probabilistic protocol that exactly implements the transpose with a single query. In contrast, we prove no-go theorems showing that neither the complex conjugate nor the adjoint can be implemented by any completely positive supermap, even probabilistically. We then overcome this impossibility by designing a virtual protocol for the complex conjugate based on quasi-probability decomposition, and show its optimality in terms of the diamond norm. As a key application, we propose a protocol to estimate the expectation values resulting from the Petz recovery map of an unknown channel, achieving an improved query complexity compared to existing methods.
Access to the complex conjugate of an unknown quantum channel is a useful resource in quantum oracle problems, motivating the question of how such access can be simulated using only a limited number of calls to the original channel. We determine the optimal deterministic protocol for approximately implementing the complex conjugate isometry V from n uses of an unknown isometry channel V:Cd→CD. We derive a closed-form expression for the optimal fidelity and prove that a parallel protocol is optimal even among general quantum superchannels, including adaptive and indefinite-causal-order strategies. The formula implies a query complexity n=Θ(d[(D−d)/ε+1]) for achieving infidelity ε. We also present a circuit construction based on the quantum Schur transform and the dual Clebsch--Gordan transform, with circuit complexity O(poly(D,1/ε)). This task is extended to the multi-copy case V⊗n↦V⊗k. For fixed d<D and k, we show that the optimal fidelity for the multi-copy case is 1−kd(D−d)/n+o(n−1), and that this value is asymptotically attained by a parallel estimation-based protocol. Finally, combining the isometry protocol with random Stinespring dilations yields a protocol for complex conjugation of unknown rank-r quantum channels whose query complexity is optimal up to a constant factor if the Kraus rank r is constant.
Unknown unitary inversion is a fundamental primitive in quantum computing and physics. Although recent work has demonstrated that quantum algorithms can invert arbitrary unknown unitaries without accessing their classical descriptions, improving the efficiency of such protocols remains an open question. In this work, we present efficient quantum algorithms for inverting unitaries with specific Hamiltonian structures, achieving significant reductions in both ancilla qubit requirements and unitary query complexity. We identify cases where unitaries encoding exponentially many parameters can be inverted using only a single query. We further extend our framework to implement unitary complex conjugation and transposition operations, and develop modified protocols capable of inverting more general classes of Hamiltonians. We have also demonstrated the efficacy and robustness of our algorithms via numerical simulations under realistic noise conditions of superconducting quantum hardware. Our results establish more efficient protocols that improve the resources required for quantum unitary inversion when prior information about the quantum system is available, and provide practical methods for implementing these operations on near-term quantum devices.
We present a quantum circuit that implements the random dilation superchannel, transforming parallel queries of an unknown quantum channel into the same number of parallel queries of a randomly chosen dilation isometry of the input channel. This is a natural generalization of the random purification channel, that transforms copies of an unknown mixed state to copies of a randomly chosen purification state. The circuit complexity of our construction is O(poly(n,logdI,logdO)), where n is the number of queries and dI and dO are the input and output dimensions of the input channel, respectively. This random dilation superchannel is extended to the sequential queries approximately, by transforming the parallel random dilation isometry into sequential random dilation unitaries with O(poly(dI)) overhead in the number of queries. We also show that our results can be further extended to the case of quantum superchannels. On the other hand, we show a no-go theorem on the exact random dilation of sequential queries with o(poly(min{dI,dO})) query overhead, showcasing a fundamental difference between the parallel and sequential cases. As an application, we show an efficient storage-and-retrieval of an unknown quantum channel, which improves the program cost exponentially in the retrieval error ε. For the case where the Kraus rank r is the least possible (i.e., r=dI/dO), we show quantum circuits that transform n parallel queries of an unknown quantum channel Λ to Θ(nα) parallel queries of Λ for any α<2 approximately, and implement its Petz recovery map for the maximally mixed reference state probabilistically and exactly.
We obtain a lower bound on the maximum number of qubits, Qn, ε(N), which can be transmitted over n uses of a quantum channel N, for a given non-zero error threshold ε. To obtain our result, we first derive a bound on the one-shot entanglement transmission capacity of the channel, and then compute its asymptotic expansion up to the second order. In our method to prove this achievability bound, the decoding map, used by the receiver on the output of the channel, is chosen to be the Petz recovery map (also known as the transpose channel). Our result, in particular, shows that this choice of the decoder can be used to establish the coherent information as an achievable rate for quantum information transmission. Applying our achievability bound to the 50-50 erasure channel (which has zero quantum capacity), we find that there is a sharp error threshold above which Qn, ε(N) scales as n.
The transpose map is the canonical nonphysical map in quantum information that is central to entanglement tests and to converse bounds for quantum capacity. It is a longstanding conjecture that transposition satisfies a dimension-independent strict submultiplicativity for physically relevant channel differences. We answer this affirmatively by proving that for every quantum channel T on L(Cd), ∥Θ∘(id−T)∥⋄≤21∥Θ∥⋄∥id−T∥⋄. The same bound holds for all Hermiticity-preserving trace-annihilating maps. Because a trace-preserving channel produces a traceless deviation from the identity, transposition cannot act as strongly on such physically generated errors as it can on arbitrary perturbations. Thus, although transposition is extremal in the unrestricted diamond-norm geometry, physical constraints mandate a universal margin in the submultiplicative bound. As one immediate consequence, the finite-error Holevo--Werner converse extends from decoding errors ε<1/2 to ε<1/2. The same tracelessness mechanism also yields rank-sensitive continuity refinements for negativity and logarithmic negativity, demonstrating its potential for broader application in quantum information.
The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This Letter sets out to rectify this lack: Using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm's usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.