Quantum Information and the PCP Theorem
Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to rea...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Raz, Ran [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2007 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC 2007 |
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| Übergeordnetes Werk: |
Enthalten in: Algorithmica - Springer-Verlag, 1986, 55(2007), 3 vom: 18. Sept., Seite 462-489 |
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| Übergeordnetes Werk: |
volume:55 ; year:2007 ; number:3 ; day:18 ; month:09 ; pages:462-489 |
| Links: |
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| DOI / URN: |
10.1007/s00453-007-9033-6 |
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| Katalog-ID: |
OLC2054841188 |
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| 520 | |a Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. | ||
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| 650 | 4 | |a Quantum interactive proofs | |
| 650 | 4 | |a Quantum advice | |
| 650 | 4 | |a Probabilistically checkable proofs | |
| 650 | 4 | |a Low degree test | |
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10.1007/s00453-007-9033-6 doi (DE-627)OLC2054841188 (DE-He213)s00453-007-9033-6-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Raz, Ran verfasserin aut Quantum Information and the PCP Theorem 2007 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2007 Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. Quantum computation Quantum information Quantum interactive proofs Quantum advice Probabilistically checkable proofs Low degree test Enthalten in Algorithmica Springer-Verlag, 1986 55(2007), 3 vom: 18. Sept., Seite 462-489 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:55 year:2007 number:3 day:18 month:09 pages:462-489 https://doi.org/10.1007/s00453-007-9033-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_65 GBV_ILN_70 GBV_ILN_105 GBV_ILN_130 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2045 GBV_ILN_2190 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4324 AR 55 2007 3 18 09 462-489 |
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10.1007/s00453-007-9033-6 doi (DE-627)OLC2054841188 (DE-He213)s00453-007-9033-6-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Raz, Ran verfasserin aut Quantum Information and the PCP Theorem 2007 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2007 Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. Quantum computation Quantum information Quantum interactive proofs Quantum advice Probabilistically checkable proofs Low degree test Enthalten in Algorithmica Springer-Verlag, 1986 55(2007), 3 vom: 18. Sept., Seite 462-489 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:55 year:2007 number:3 day:18 month:09 pages:462-489 https://doi.org/10.1007/s00453-007-9033-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_65 GBV_ILN_70 GBV_ILN_105 GBV_ILN_130 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2045 GBV_ILN_2190 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4324 AR 55 2007 3 18 09 462-489 |
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10.1007/s00453-007-9033-6 doi (DE-627)OLC2054841188 (DE-He213)s00453-007-9033-6-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Raz, Ran verfasserin aut Quantum Information and the PCP Theorem 2007 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2007 Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. Quantum computation Quantum information Quantum interactive proofs Quantum advice Probabilistically checkable proofs Low degree test Enthalten in Algorithmica Springer-Verlag, 1986 55(2007), 3 vom: 18. Sept., Seite 462-489 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:55 year:2007 number:3 day:18 month:09 pages:462-489 https://doi.org/10.1007/s00453-007-9033-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_65 GBV_ILN_70 GBV_ILN_105 GBV_ILN_130 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2045 GBV_ILN_2190 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4324 AR 55 2007 3 18 09 462-489 |
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10.1007/s00453-007-9033-6 doi (DE-627)OLC2054841188 (DE-He213)s00453-007-9033-6-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Raz, Ran verfasserin aut Quantum Information and the PCP Theorem 2007 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2007 Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. Quantum computation Quantum information Quantum interactive proofs Quantum advice Probabilistically checkable proofs Low degree test Enthalten in Algorithmica Springer-Verlag, 1986 55(2007), 3 vom: 18. Sept., Seite 462-489 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:55 year:2007 number:3 day:18 month:09 pages:462-489 https://doi.org/10.1007/s00453-007-9033-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_65 GBV_ILN_70 GBV_ILN_105 GBV_ILN_130 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2045 GBV_ILN_2190 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4324 AR 55 2007 3 18 09 462-489 |
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10.1007/s00453-007-9033-6 doi (DE-627)OLC2054841188 (DE-He213)s00453-007-9033-6-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ Raz, Ran verfasserin aut Quantum Information and the PCP Theorem 2007 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC 2007 Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. Quantum computation Quantum information Quantum interactive proofs Quantum advice Probabilistically checkable proofs Low degree test Enthalten in Algorithmica Springer-Verlag, 1986 55(2007), 3 vom: 18. Sept., Seite 462-489 (DE-627)129197564 (DE-600)53958-2 (DE-576)014456958 0178-4617 nnns volume:55 year:2007 number:3 day:18 month:09 pages:462-489 https://doi.org/10.1007/s00453-007-9033-6 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_65 GBV_ILN_70 GBV_ILN_105 GBV_ILN_130 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2045 GBV_ILN_2190 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4324 AR 55 2007 3 18 09 462-489 |
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Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. © Springer Science+Business Media, LLC 2007 |
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Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. © Springer Science+Business Media, LLC 2007 |
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Abstract Our main result is that the membership x∈SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ〉 the verifier only needs to read one block of the classical proof. This shows that if a short quantum witness is available then a (classical) PCP with only one query is possible. Our second result is that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x∈L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomial-size quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. The advice |ΨL,n〉 given to the verifier can also be replaced by a classical probabilistic advice, as long as this advice is kept as a secret from the prover. Our result can hence be interpreted as: the class IP/rpoly contains all languages. For the proof of the second result, we introduce the quantum low-degree-extension of a string of bits. The main result requires an additional machinery of quantum low-degree-test. © Springer Science+Business Media, LLC 2007 |
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