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ECS 189A Lecture 17: Quantum Circuit
创建日期：2024-05-20 15:28:35
标签： ECS 189A
Quantum Circuit

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ECS 189A Lecture 17: Quantum Circuit

Identities
ßß
ßß ßß ß
ß
1
Recap
• Familiarize yourself with circuit identities! (They are very useful.)
• Single-qubit gates: Switching between X , Y , and Z
• Controlled-gates: Think of it as a little computer program, and then
simplify.
Next lecture: More on controlled gates, and then measurements
2
Controlled-gates
H H
H H
=?
3
Controlled-gates: Propagation
X
=
U1
U2
U1,U2 =?
4
Controlled-gates: Propagation
X
=
U1
U2
U1,U2 =?
5
Controlled-gates: Propagation
Z
=
U1
U2
U1,U2 =?
6
Controlled-gates: Propagation
Z
=
U1
U2
U1,U2 =?
7
Composing two controlled-gates
=?
8
Measurement
9
Measurement
X
=
Z
H
.
What does this mean? And why is this true?
10
Measurement outcome
Throughout this course, we assumed that the measurement is performed
in the Z -basis.
Z =
(
1 0
0 −1
)
.
Depending on the measurement outcome, we get |0⟩ or |1⟩, the
eigenstates of Z .
11
Eigenstate vs. Eigenvalue
For each eigenstate, there is an associated eigenvalue:
Z |0⟩ = (+1)︸︷︷︸
Eigenvalue
|0⟩
Z |1⟩ = (−1)︸︷︷︸
Eigenvalue
|1⟩
Since the eigenvalues are different, we could have specified the eigenstate
by their eigenvalues.
12
Measurement in terms of eigenvalues
Z
The measurement device is a machine that reports one of the
eigenvalues, +1 or −1.
• +1 case: |0⟩ (+1 eigenstate)
• −1 case: |1⟩ (−1 eigenstate)
Let |ψ⟩ be the state.
Prob[Measured + 1] = |⟨0|ψ⟩|2
Prob[Measured − 1] = |⟨1|ψ⟩|2
13
Measurement in terms of eigenvalues
Z
The measurement device is a machine that reports one of the
eigenvalues, +1 or −1.
• +1 case: |0⟩ (+1 eigenstate)
• −1 case: |1⟩ (−1 eigenstate)
Let |ψ⟩ be the state.
Prob[Measured + 1] = |⟨0|ψ⟩|2
Prob[Measured − 1] = |⟨1|ψ⟩|2
13
Measurement in terms of eigenvalues
X
The measurement device is a machine that reports one of the
eigenvalues, +1 or −1.
• +1 case: |+⟩ (+1 eigenstate)
• −1 case: |−⟩ (−1 eigenstate)
Let |ψ⟩ be the state.
Prob[Measured + 1] = |⟨+|ψ⟩|2
Prob[Measured − 1] = |⟨−|ψ⟩|2
14
Measurement in terms of eigenvalues
X
The measurement device is a machine that reports one of the
eigenvalues, +1 or −1.
• +1 case: |+⟩ (+1 eigenstate)
• −1 case: |−⟩ (−1 eigenstate)
Let |ψ⟩ be the state.
Prob[Measured + 1] = |⟨+|ψ⟩|2
Prob[Measured − 1] = |⟨−|ψ⟩|2
14
Proving the identity
X
=
Z
H
.
15
Exercise
Y
=
X
U
.
What is U?
16
Measurement and feedback
17
Recap
• Controlled-unitaries
• Measurements in different bases
Congrats, now you learned everything about quantum circuits!
Next week: Quantum computation, in the language of quantum circuit
(ex. Deutsch-Jozsa algorithm)
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MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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101H1 IB9520 FINM 3008 ECON7310 FIT 1006 MANAGEMENT MATH3821 MTHM504 CS975 COMM1100 ELEC2133 MMAI 5500 MIS611 ELEC3114 CS 173 ELEC 3105 ISO 9001 IDSC10 Rural livelihoods and globalisati IDSC10 K17 202 CS823 MAST90007 SPG8024 MAT246 EG-264 CVEN2303 INFS1603 3PR3 MN-2508 4CCE1MCP SPG8030 CMT224 Inflation consumption Applying MAT137 CSE 111 DPBS1110 FINS5530 ELEC4613 GSBS6200 COMP4336 Math W54 DIFFERENTIAL 7PAVMALM ACC 1110 ME 3210 MATH2801 MGMT20001 ECON 6202 PSTAT 120B COMP9024 ENG5027 comp3219 ELE00107M ECON20003 ECON2103 FINA2209 MAST20018 LUBS5308M01 MATH2023 Computation COMP1600 CHEM 3X23 BBMM506 FNCE20003 BMET2922 BISM7216 financial ECMT5001 ACST3059 MAST30012 COMP3511 ECOS 2201 MKTG7512 ECON2540 MTH1010 MAT1841 COMP4600 ECON7740 PAPER C331 CAP 6617 ECON2420 ACCT7107 HLTH 7001 ECOS3002 CVEN30010 MAST30021 MATH1014 BIOL1007 ECOS 3010 SWEN90016 BISM7202 ACCT90012 INFO3315 DSCI 553 INFO90002 COMP3419 CITS 3001 EE5731 FINA3326 NFS 284 COMM2501 FIT2104 PHYS3035 RSM260 MAN3080 CHEM30028 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