3 Search Results for "Reichardt, Ben W."

Document
DOI: 10.4230/LIPIcs.CCC.2024.30
The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise

Authors: Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
The class MIP^* of quantum multiprover interactive proof systems with entanglement is much more powerful than its classical counterpart MIP [Babai et al., 1991; Zhengfeng Ji et al., 2020; Zhengfeng Ji et al., 2020]: while MIP = NEXP, the quantum class MIP^* is equal to RE, a class including the halting problem. This is because the provers in MIP^* can share unbounded quantum entanglement. However, recent works [Qin and Yao, 2021; Qin and Yao, 2023] have shown that this advantage is significantly reduced if the provers' shared state contains noise. This paper attempts to exactly characterize the effect of noise on the computational power of quantum multiprover interactive proof systems. We investigate the quantum two-prover one-round interactive system MIP^*[poly,O(1)], where the verifier sends polynomially many bits to the provers and the provers send back constantly many bits. We show noise completely destroys the computational advantage given by shared entanglement in this model. Specifically, we show that if the provers are allowed to share arbitrarily many EPR states, where each EPR state is affected by an arbitrarily small constant amount of noise, the resulting complexity class is equivalent to NEXP = MIP. This improves significantly on the previous best-known bound of NEEEXP (nondeterministic triply exponential time) [Qin and Yao, 2021]. We also show that this collapse in power is due to the noise, rather than the O(1) answer size, by showing that allowing for noiseless EPR states gives the class the full power of RE = MIP^*[poly, poly]. Along the way, we develop two technical tools of independent interest. First, we give a new, deterministic tester for the positivity of an exponentially large matrix, provided it has a low-degree Fourier decomposition in terms of Pauli matrices. Secondly, we develop a new invariance principle for smooth matrix functions having bounded third-order Fréchet derivatives or which are Lipschitz continuous.
Cite as

Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 30:1-30:71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dong_et_al:LIPIcs.CCC.2024.30,
  author =	{Dong, Yangjing and Fu, Honghao and Natarajan, Anand and Qin, Minglong and Xu, Haochen and Yao, Penghui},
  title =	{{The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{30:1--30:71},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.30},
  URN =		{urn:nbn:de:0030-drops-204263},
  doi =		{10.4230/LIPIcs.CCC.2024.30},
  annote =	{Keywords: Interactive proofs, Quantum complexity theory, Quantum entanglement, Fourier analysis, Matrix analysis, Invariance principle, Derandomization, PCP, Locally testable code, Positivity testing}
}
Document
DOI: 10.4230/LIPIcs.TQC.2021.5
Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits

Authors: Prithviraj Prabhu and Ben W. Reichardt

Published in: LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)

Abstract
We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For fault-tolerant syndrome extraction, we show an exponential reduction in qubit overhead over the previous best protocol. For a weight-w stabilizer, we demonstrate that stabilizer measurement tolerating one fault (distance-three) needs at most ⌈ log₂ w ⌉ + 1 ancillas. If qubits reset quickly, four ancillas suffice. We also study the preparation of cat states, simple yet versatile entangled states. We prove that the overhead needed for distance-three fault tolerance is only logarithmic in the cat state size. These results could be useful both for near-term experiments with a few qubits, and for the general study of the asymptotic resource requirements of syndrome measurement and state preparation. For 'a' measured flag bits, there are 2^a possible flag patterns that can identify faults. Hence our results come from solving a combinatorial problem: the construction of maximal-length paths in the a-dimensional hypercube, corresponding to maximal-weight stabilizers or maximal-weight cat states.
Cite as

Prithviraj Prabhu and Ben W. Reichardt. Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{prabhu_et_al:LIPIcs.TQC.2021.5,
  author =	{Prabhu, Prithviraj and Reichardt, Ben W.},
  title =	{{Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits}},
  booktitle =	{16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)},
  pages =	{5:1--5:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-198-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{197},
  editor =	{Hsieh, Min-Hsiu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.5},
  URN =		{urn:nbn:de:0030-drops-140001},
  doi =		{10.4230/LIPIcs.TQC.2021.5},
  annote =	{Keywords: Quantum error correction, fault tolerance, quantum state preparation, combinatorics}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.48
Overlapping Qubits

Authors: Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
An ideal system of n qubits has 2^n dimensions. This exponential grants power, but also hinders characterizing the system's state and dynamics. We study a new problem: the qubits in a physical system might not be independent. They can "overlap," in the sense that an operation on one qubit slightly affects the others. We show that allowing for slight overlaps, n qubits can fit in just polynomially many dimensions. (Defined in a natural way, all pairwise overlaps can be <= epsilon in n^{O(1/epsilon^2)} dimensions.) Thus, even before considering issues like noise, a real system of n qubits might inherently lack any potential for exponential power. On the other hand, we also provide an efficient test to certify exponential dimensionality. Unfortunately, the test is sensitive to noise. It is important to devise more robust tests on the arrangements of qubits in quantum devices.
Cite as

Rui Chao, Ben W. Reichardt, Chris Sutherland, and Thomas Vidick. Overlapping Qubits. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 48:1-48:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chao_et_al:LIPIcs.ITCS.2017.48,
  author =	{Chao, Rui and Reichardt, Ben W. and Sutherland, Chris and Vidick, Thomas},
  title =	{{Overlapping Qubits}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{48:1--48:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.48},
  URN =		{urn:nbn:de:0030-drops-81826},
  doi =		{10.4230/LIPIcs.ITCS.2017.48},
  annote =	{Keywords: Quantum computing, Qubits, Dimension test}
}
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