ECCC-Report TR10-057https://eccc.weizmann.ac.il/report/2010/057Comments and Revisions published for TR10-057en-usTue, 22 Oct 2013 22:46:53 +0300
Revision 3
| A Full Characterization of Quantum Advice |
Scott Aaronson,
Andrew Drucker
https://eccc.weizmann.ac.il/report/2010/057#revision3We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.Tue, 22 Oct 2013 22:46:53 +0300https://eccc.weizmann.ac.il/report/2010/057#revision3
Revision 2
| A Full Characterization of Quantum Advice |
Scott Aaronson,
Andrew Drucker
https://eccc.weizmann.ac.il/report/2010/057#revision2We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.Fri, 25 Nov 2011 19:14:52 +0200https://eccc.weizmann.ac.il/report/2010/057#revision2
Revision 1
| A Full Characterization of Quantum Advice |
Scott Aaronson,
Andrew Drucker
https://eccc.weizmann.ac.il/report/2010/057#revision1We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.Fri, 02 Apr 2010 18:19:20 +0300https://eccc.weizmann.ac.il/report/2010/057#revision1
Paper TR10-057
| A Full Characterization of Quantum Advice |
Scott Aaronson,
Andrew Drucker
https://eccc.weizmann.ac.il/report/2010/057We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.Thu, 01 Apr 2010 23:46:37 +0300https://eccc.weizmann.ac.il/report/2010/057