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arXiv:2512.14588

Sequential realization of Quantum Instruments

Soham Sau, Michal Sedlák

quant-phSubmitted Dec 16, 2025• Announced Dec 17, 2025• 0 citations
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Abstract

In adaptive quantum circuits classical results of mid-circuit measurements determine the upcoming gates. This allows POVMs, quantum channels or more generally quantum instruments to be implemented sequentially, so that fewer qubits need to be used at each of the $N$ measurement steps. In this paper, we mathematically describe these problems via adaptive sequence of instruments (ASI) and show how any instrument can be decomposed into it. Number of steps $N$ and number of ancillary qubits $n_A$ needed for actual implementation are crucial parameters of any such ASI. We show an achievable lower bound on the product $N.n_A$ and we determine in which situations this tradeoff is likely to be optimal. Contrary to common intuition we show that for quantum instruments which transform $n$ to $m(>n)$ qubits, there exist $N$-step ASI implementing them just with $(m-n)$ ancillary qubits, which are remeasured $(N-1)$ times and finally used as output qubits.

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25 pages, 4 figures

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In adaptive quantum circuits classical results of mid-circuit measurements determine the upcoming gates. This allows POVMs, quantum channels or more generally quantum instruments to be implemented sequentially, so that fewer qubits need to be used at each of the NNN measurement steps. In this paper, we mathematically describe these problems via adaptive sequence of instruments (ASI) and show how any instrument can be decomposed into it. Number of steps NNN and number of ancillary qubits nAn_AnA​ needed for actual implementation are crucial parameters of any such ASI. We show an achievable lower bound on the product N.nAN.n_AN.nA​ and we determine in which situations this tradeoff is likely to be optimal. Contrary to common intuition we show that for quantum instruments which transform nnn to m(>n)m(>n)m(>n) qubits, there exist NNN-step ASI implementing them just with (m−n)(m-n)(m−n) ancillary qubits, which are remeasured (N−1)(N-1)(N−1) times and finally used as output qubits.
Scaling up the number of qubits available on quantum processors remains technically demanding even in the long term; it is therefore crucial to clarify the number of qubits required to implement a given quantum operation. For the most general class of quantum operations, known as quantum instruments, the qubit requirements are not well understood, especially when mid-circuit measurements and delayed input preparation are permitted. In this work, we characterize lower and upper bounds on the number of qubits required to implement a given quantum instrument in terms of the causal structure of the instrument. We further apply our results to entanglement distillation protocols based on stabilizer codes and show that, in these cases, the lower and upper bounds coincide, so the optimal qubit requirement is determined. In particular, we compute that the optimal number of qubits is 3 for the [[9,1,3]][[9,1,3]][[9,1,3]]-code-based protocol and 4 for the [[5,1,3]][[5,1,3]][[5,1,3]]-code-based protocol.
Existing quantum systems provide very limited physical qubit counts, trying to execute a quantum algorithm/circuit on them that have a higher number of logical qubits than physically available lead to a compile-time error. Given that it is unrealistic to expect existing quantum systems to provide, in near future, sufficient number of qubits that can accommodate large circuit, there is a pressing need to explore strategies that can somehow execute large circuits on small systems. In this paper, first, we perform an analysis to identify the qubits that are most suitable for circuit resizing. Our results reveal that, in most quantum programs, there exist qubits that can be reused mid-program to serially/sequentially execute the circuit employing fewer qubits. Motivated by this observation, we design, implement and evaluate a compiler-based approach that i) identifies the qubits that can be most beneficial for serial circuit execution; ii) selects those qubits to reuse at each step of execution for size minimization of the circuit; and iii) minimizes Middle Measurement (MM) delays due to impractical implementation of shots to improve the circuit reliability. Furthermore, since our approach intends to execute the circuits sequentially, the crosstalk errors can also be optimized as a result of the reduced number of concurrent gates. The experimental results indicate that our proposed approach can (i) execute large circuits that initially cannot fit into small circuits, on small quantum hardware, and (ii) can significantly improve the PST of the results by 2.1X when both original and our serialized programs can fit into the target quantum hardware.
We study the implementation of quantum channels with quantum computers while minimizing the experimental cost, measured in terms of the number of controlled-not (cnot) gates required (single-qubit gates are free). We consider three different models. In the first, the quantum circuit model (QCM), we consider sequences of single-qubit and cnot gates and allow qubits to be traced out at the end of the gate sequence. In the second (RandomQCM), we also allow external classical randomness. In the third (MeasuredQCM) we also allow measurements followed by operations that are classically controlled on the outcomes. We prove lower bounds on the number of cnot gates required and give near-optimal decompositions in almost all cases. Our main result is a MeasuredQCM circuit for any channel from mmm qubits to nnn qubits that uses at most one ancilla and has a low cnot count. We give explicit examples for small numbers of qubits that provide the lowest known cnot counts.
Because a quantum measurement generally disturbs the state of a quantum system, one might think that it should not be possible for a sender and receiver to communicate reliably when the receiver performs a large number of sequential measurements to determine the message of the sender. We show here that this intuition is not true, by demonstrating that a sequential decoding strategy works well even in the most general ‘one-shot’ regime, where we are given a single instance of a channel and wish to determine the maximal number of bits that can be communicated up to a small failure probability. This result follows by generalizing a non-commutative union bound to apply for a sequence of general measurements. We also demonstrate two ways in which a receiver can recover a state close to the original state after it has been decoded by a sequence of measurements that each succeed with high probability. The second of these methods will be useful in realizing an efficient decoder for fully quantum polar codes, should a method ever be found to realize an efficient decoder for classical-quantum polar codes.
Quantum Random Access Codes (QRACs) are key tools for a variety of protocols in quantum information theory. These are commonly studied in prepare-and-measure scenarios in which a sender prepares states and a receiver measures them. Here, we consider a three-party prepare-transform-measure scenario in which the simplest QRAC is implemented twice in sequence based on the same physical system. We derive optimal trade-off relations between the two QRACs. We apply our results to construct semi-device independent self-tests of quantum instruments, i.e. measurement channels with both a classical and quantum output. Finally, we show how sequential QRACs enable inference of upper and lower bounds on the sharpness parameter of a quantum instrument.
Quantum Random Access Codes (QRACs) are key tools for a variety of protocols in quantum information theory. These are commonly studied in prepare-and-measure scenarios in which a sender prepares states and a receiver measures them. Here, we consider a three-party prepare-transform-measure scenario in which the simplest QRAC is implemented twice in sequence based on the same physical system. We derive optimal trade-off relations between the two QRACs. We apply our results to construct semi-device independent self-tests of quantum instruments, i.e. measurement channels with both a classical and quantum output. Finally, we show how sequential QRACs enable inference of upper and lower bounds on the sharpness parameter of a quantum instrument.