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# Quantum advantage for computations with limited space

## Abstract

Quantum computers promise the ability to solve problems that are intractable in the classical setting1, but in many cases this is not rigorously proven. It is often possible to establish a provable theoretical advantage for quantum computations by restricting the computational power2,3,4,5,6,7,8. In multiple cases, quantum advantage over these restricted models was demonstrated experimentally9,10,11,12. Here we consider space-restricted computations that use only one computational classical or quantum bit with a read-only memory as input. We show that n-bit symmetric Boolean functions can be implemented exactly in this framework through the use of quantum signal processing13 and O(n2) gates, but in the analogous classical computations some of the functions may only be evaluated with probability $$1/2 + O\left(n/{\sqrt{2}}^n\right)$$. We experimentally demonstrate computations of three-bit to six-bit symmetric Boolean functions by quantum circuits with an algorithmic success probability that exceeds the classical limit. This shows that in computations, quantum scrap space offers an advantage over analogous classical space, and calls for an in-depth exploration of space–time trade-offs in quantum circuits.

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## Data availability

All relevant data are available from the corresponding author upon request.

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## Acknowledgements

We thank N. Kanazawa and E. Chen for experimental contributions and J. M. Gambetta for discussions. S.B. and T.J.Y. are partially supported by the IBM Research Frontiers Institute.

## Author information

Authors

### Contributions

D.M. designed the research, and S.B., D.M. and T.J.Y. developed the theory. J.-S.K. and S.S. ran the experiments. All authors contributed to writing the manuscript.

### Corresponding author

Correspondence to Dmitri Maslov.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Connectivity diagram of ibmq_berlin.

Connectivity diagram of ibmq_berlin with the qubits used in the experiment highlighted.

### Extended Data Fig. 2 Pulse sequence and calibration.

a, Pulse diagram of the echoed CR sequence including the rotary echoes applied to the target qubit. The sampling time is 0.2222 ns per sample. ‘d10’ and ‘d12’ denote the drive channels for qubits Q10 and Q12 respectively, while ‘u21’ denotes the cross-resonance channel for the control qubit, 10. b, Fine amplitude calibration of the echoed ZXπ/4 CR pulse sequence for qubits Q15 and Q12. Initially the ZXπ/4 pulses are applied in repetitions of 2 to ensure a full rotation about the Bloch sphere. At 16 repetitions, the pulses are applied in repetitions of 4 to apply π pulses about the Bloch sphere equator in order to amplify amplitude errors. c, Rabi oscillations of the target qubit used to calibrate a 2π rotary echo for Q12.

### Extended Data Fig. 3 Randomized benchmarking for decay curves characterizing the controlled-Rx(π/2) calibrated in Qiskit Pulse.

Pair (Q10, Q12) is shown in (a), (Q15, Q12) in (b), and (Q13, Q12) in (c). Data is shown in red and the fit is shown in blue. Error per Clifford (EPC) is shown in each plot and the error per controlled-Rx(π/2) is taken to be $$\frac{1}{3}$$EPC.

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Maslov, D., Kim, JS., Bravyi, S. et al. Quantum advantage for computations with limited space. Nat. Phys. 17, 894–897 (2021). https://doi.org/10.1038/s41567-021-01271-7

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