**Michael A. Nielsen and Isaac L. Chuang**

Thanks to all the people who've contributed to this errata list: Scott Aaronson, Marcus Curty Alonso, Andris Ambainis, Dave Bacon, Mahesh Bandi, Charles Bennett, Mike Brown, Todd Brun, Mark Byrd, Jay Cheng, Andrew Childs, Richard Cleve, Mauro D'Ariano, Jennifer Dodd, Andrew Duggins, Michael Gagen, Ian Glendinning, Ben Gold, Jeroen van de Graaf, Bob Griffiths, Michael Hall, Charles Hill, Gert Ingold, Lawrence Ioannou, Matthew Paul Johnson, Alexei Kaltchenko, Phil Kaye, Yeong Cherng Liang, Naile Liu, Tom Luu, Mark Madden, Ryutaroh Matsumoto, David Meyer, John Obrecht, Ivan Oliveira, Christina Pencarski, Damian Pope, Terry Rudolph, Andy Shiekh, Peter Shor, Alessandro de Sousa Villar, Damonn Sypher, Rob Thew, Julian Ting, Lieven Vandersypen, Xiangbin Wang, and Qianchuan Zhao.

Printing History

The book is now in its fifth printing. Note that the second, fourth and fifth printings contain corrections based on the errata below. The third printing does not contain any corrections beyond those in the second printing.

First, Second, Third, Fourth & Fifth Printing (July, 2002)

- pp 4
- On page 4, in the first full sentence on the page,
there is a missing comma after ``qubits''.
- pp 6
- In the errata for the First
through Fourth printings there is an error related to page 6, and the
Solovay-Strassen test. Namely, ``no deterministic test'' should be
``no efficient deterministic test''.
- pp 19
- The caption for Figure 1.5 should mention that
is shorthand for the logical negation of .
- pp 90
- Just before Exercise 2.59, ``The following two exercises
[...]'' should be ``The following three exercises [...]''.
- pp 95
- Immediately after Exercise 2.67, ``Next, suppose we
perform ...'' should be replaced by ``After letting act on
, suppose we perform ....
- pp 95
- On the left-hand side of Eq. (2.130), in the denominator
should be
, and
should be
.
- pp 141
- In Exercise 3.16, should be .
- pp 144
- In Exercise 3.19, the time to solve REACHABILITY ought
to be , and the time to decide whether a graph is connected .
- pp 150
- In the statement of the graph isomorphism problem, at
the top of the page, the final in the statement of the problem
ought to be .
- Sec 3.2.5
- An important lemma is missing from the section. We
explain in this section how, given a classical circuit to compute a
function it is possible to produce a reversible circuit using
comparably many reversible gates to compute the transformation
. (The proof is given explicitly for
functions having a single bit, zero or one, as their output, but
obviously holds also for functions with multi-bit outputs, with
denoting bitwise modulo two addition in that case.) The
missing lemma is that if and are permutations which
*both*have polynomial-size classical circuits, possibly not reversible, then there is a polynomial size reversible circuit to compute , with no ancilla bits. The proof is to compute , using the already-introduced techniques of reversible computation. This result is implicitly used, for example, in the chapter on Shor's algorithm. - pp 195-196
- The argument leading to Equation (4.81) needs some
modification, in the light of the earlier correction to Exercise 4.11,
on page 176. We sketch the modification here. The key is that
according to the modified exercise, an arbitrary unitary can be
written as a finite product of rotations about the and axes, up to an unimportant global phase. In particular, for fixed
and the number of rotations required is a constant,
, independent of . Modifying Equation (4.80) and
Equation (4.81) appropriately we see that the error in the
approximation is
. We could also have modified the
choice of , earlier, so that the right-hand side of Equation (4.76)
becomes , in which case the bound on the right-hand
side of (4.79) would have become and that on the
right-hand side of (4.81) would have remained .
- pp 177
- In Equations (4.20) and (4.22) the minus sign on the
right-hand side should be a plus.
- pp 203
- Theorem 5.3 should be amended so that the lower bound in
Eq. (5.60) is . See the discussion below for page 634.
Note that, so far as we can see, this error does not propagate to later
discussions of the factoring algorithm. (Notifications of places where it
does propagate would be welcome!)
- pp 207
- The reasoning at the end of the proof, after
Eq. (4.100), is incorrect. In fact, a conceptually much simpler (and
correct) argument may substitue. From Eq. (4.100) we see that
. Raising this to the
th power gives
, from which the result follows. Note that to
understand how the error term scales when raised to a power, we must
use an inequality like Eq. (4.69), in Box 4.1, on page 195.
- pp 208
- In ``
**Inputs**'', the label ``(3)'' is repeated. The second one should be ``(4)''. - pp 234
- In the exercise on factoring , the final should
be , not .
- pp 238
- In Equation (5.69) the square root should be removed from
over .
- pp 285
- In Box 7.2, just after Eq. (7.8) there is an integral missing
a .
- pp 285
- In the third last line of Box 7.2,
should be
.
- pp 286
- Just after Equation (7.15), ``
'' should
be ``''..
- pp 298
- After Equation (7.56), and should be described
as annihilation and creation operators, not creation and annihilation
operators.
- pp 301
- In the third-last line of the first paragraph of
Box 7.6, should be .
- pp 326
- In the second line of the ``Single spin dynamics''
section there is a missing vector notation on ``
''.
- pp 373
- On the first and second line, the sum
``
'' should only be over , not and .
- pp 392
- After Eq. (8.168) there are two places where
should be .
- pp 411
- In Equation (9.73), the should be .
Equation (9.69) on the same page needs to be modified accordingly,
with
inside the trace becoming , and
similarly on the line immediately below needs to become
.
- 450
- On the third line prior to Equation (10.66),
``
'' should be ``
''.
- pp 464
- We state that measurement of observables in the Pauli
group may be simulated using operations, provided one is using
the stabilizer formalism to describe the quantum state of the system.
This is true, but not obvious. The discussion in the book
makes clear how the simulation may be performed in time. Scott
Aaronson (private communication) has shown how this may be improved to
time.
- pp 481
- Just before Equation (10.114), ``Note that we have''
should be replaced by ``Choosing the smallest satisfying (10.113),
so that the inequality in (10.113) is close to being saturated, and
rearranging (10.113), we have:'' The equation (10.114) should then be
changed so that the first equality sign is replaced by an sign.
- pp 487
- Exercise 10.66 has an error. The relation
should read
.
- pp 521
- In Equation (11.104) there is a that should
be .
- pp 544
- Just after Equation (12.49) there is a
that
should be
. In the couple of lines following this,
dedicated parentheses watchers will find two missing right parentheses.
- pp 629
- In Equation (A.4.12) and in the line immediately following,
``37'' should be ``43''.
- pp 634
- In the statement of Theorem A4.13, the probability in
Eq. (A4.32) should be bounded below by , not .
The proof of the theorem also needs some modification, which we sketch
here. The proof as written shows that all the must take the
same value in order to have odd or
. This
part of the proof is correct. The error is to assume that this
implies, based on Lemma A4.12, that the probability that is odd or
is at most . In fact, there is no
constraint on implied by Lemma A4.12, it is only
that are constrained to be equal to , each
contributing a factor to the decrease in probability, by
Lemma A4.12. As a result, Eq. (A4.33) should also be amended so the
right-hand side is .
- pp 638
- In Equation (A4.55), should be .
- pp 661
- In the Bilbiography entry for [Sho96], Shor's paper on
fault-tolerant quantum computing, the symposium name ``Fundamentals of
Computer Science'' should be ``Foundations of Computer Science''.
- pp 663
- In the Bibliography entry for [VR89] the correct
journal informamtion is
*Phys. Rev. A*, 40 (5): 2847-2849, 1989.

First, Second, Third & Fourth Printing (October, 2001)

- pp 2
- One the first line of Section 1.1.1, ``a unheralded'' should
be ``an unheralded''.
- pp 4
- On line 2, ``qubits'' should be ``quantum bits (or
*qubits*)''. - pp 6
- The sentence beginning ``Of especial interest at the time
the Solovay-Strassen...'' is somehwat ambiguous. It should be
replaced by: ``The Solovay-Strassen test was of especial significance
at the time it was proposed as no deterministic test for primality was
then known, nor is one known at the time of this writing.''
- pp 12
- In the first line of Section 1.1.2, ``some at the
history'' should read ``some of the history''.
- pp 19
- The geometric description of the Hadamard gate is
incorrect. To correct the description, note that a Hadamard gate
corresponds to a rotation of the Bloch sphere by 90 degrees about the
axis, followed by a degree rotation about the
axis. Equivalently, it can be described by a 180 degree rotation
about the axis
.
- pp 61
- On the second line of the second paragraph,
``but rather the rather'' should be ``but rather the''.
- pp 88
- Four lines below Eq(2.115), the formula in the text
should read
.
- pp 105
- In fixing Exercise 2.73 for the fourth printing, we
inadvertently took out the definition of the support for a Hermitian
operator. The exercise should be amended to include this definition:
``The
*support*of a Hermitian operator is the vector space spanned by the eigenvectors of with non-zero eigenvalues.'' - pp 169
- Fourth line from top: should be .
- pp 205
- The first differential in Eq. (4.88) should be with
respect to time , not to .
- pp 218
- Equation (5.9) should have a base of instead of for
the exponential multiplying .
- pp 254
- Line 3 of the quantum search algorithm has a superscript
which should be just , denoting application of the
operator times.
- pp 276
- The [Zal98] should instead refer to [Zal99],
C. Zalka. Grover's quantum searching algorithm is optimal. Physical
review A, 60:2746-2751, 1999.
- pp 321
- ``basic'' is misspelled in the caption of Fig 7.12
- pp 323
- End of second paragraph: ``between'' is misspelled.
- pp 330-331
- In (7.144), should be replaced by .
And at the end of the first paragraph on page 331, the exponent is
missing a minus sign, and should read
.
- pp 363
- In (8.27), should be replaced by and
by .
- pp 382
- In the last line of Exercise 8.25 should read
.
- pp 386
- In Exercise 8.30, should be replaced by ,
so that it reads as follows:
**Exercise 8.30: ()**The phase coherence relaxation rate is just the exponential decay rate of the off-diagonal elements in the qubit density matrix, while is the decay rate of the diagonal elements (see Equation (7.144)). Amplitude damping has*both*nonzero and rates; show that for amplitude damping . Also show that if amplitude and phase damping are*both*applied then . - pp 412
- In Equation (9.75) and subsequently an unexplained
``'' suddenly appears in the proof. This arises from the polar
decomposition,
.
- pp 418
- In Exercise 9.22 we need to add the assumption that the
metric be unitarily invariant, that is,
for all density matrices and
and unitary matrices .
- pp 423
- An unfortunate transposition occurred during final
typesetting of the book: the square root appearing on the right-hand
side of (9.144) should be removed, and placed instead over the
quantity on the right-hand side of (9.145).
- pp 423
- In the statement of Problem 9.2 ``Show that there is a
set [...]'' should be replaced by ``Show that for each there is
a set [...]''.
- pp 424
- The ``History and further reading'' section should
include a citation to C. A. Fuchs and J. van de Graaf, ``Cryptographic
distinguishability measures for quantum-mechanical states'', IEEE
Transactions on Information Theory
**45**(4), 1216-1227 (1999). This paper is the origin of the inequality (9.110), and is also a good overview of distance measures for quantum information, especially in the context of quantum cryptography. - pp 449
- Eq.(10.63) should have instead of .
- pp 532
- In the second-last line of Box 12.1, ``mean'' should be
``meant''.
- pp 600
- In equation (12.206) the two instances of on the
r.h.s. should be and , respectively. There should also be
a sum over .
- pp 654
- The citation for [EJ96] states that the article in
question begins on page 1, when it actually begins on page 733.

First & Second Printing (January, 2001)

- pp 20
- Equation (1.17) contains an extra comma between the
second and third matrices on the right hand side.
- pp 36
- In step 3 of the Deutsch-Jozsa algorithm, a
normalization factor of is missing.
- pp 36
- In step 4 of the Deutsch-Jozsa algorithm, the
normalization factor in the denominator should be rather than
.
- pp 48
- On line 8, ``procotols'' should be ``protocols''.
- pp 88
- In Equation (2.114), should be .
- pp 105
- Michael Hall has pointed out that Exercise (2.73) needs
to be replaced, as follows:
**Exercise 2.73:**Let be a density operator. A*minimal ensemble*for is an ensemble containing a number of elements equal to the rank of . Let be any state in the support of . Show that there is a minimal ensemble for that contains , and moreover that in any such ensemble must appear with probability

where is defined to be the inverse of , when is considered as an operator acting only on the support of . (This definition removes the problem that may not have an inverse.) - pp 106
- Equation (2.189) should read:

(2.189) - pp 214-215
- The ``History and further reading'' section should
acknowledge that the suggestion (made in Section 4.6) that it may be
possible to use non-computational basis starting states to obtain
computational power beyond the quantum circuits model was made by
Daniel Gottesman and Michael Nielsen (private communication).
- pp 225
- In the line beginning
**Inputs:**, ``wich'' should be ``which''. - pp 267
- On the first line of the last paragraph ``the principle
use'' should be ``the principal use''.
- pp 280
- Just after Equation(7.1) it is stated that
. The correct expression is
.
- pp 291-292
- At the bottom of p291, is defined incorrectly;
it should be
(to be consistent with
(7.24)). This change introduces a minuns sign which propagates
through (7.33), giving the new text and equations:
Since it follows from and that and , for , we obtain for the expansion of the series coefficients , , , , which in general are

(7.28) (7.29) From this, our desired result follows straightforwardly:

(7.30) (7.31) (7.32) (7.33)

- pp 303
- In Equation (7.78), and should be interchanged.
- pp 309
- The static potential on the third last line of the page
should read ``
''. That is, the factor is now
inside the square brackets.
- pp 316
- In Equation (7.110) the term
appearing in the argument of should not be squared.
- pp 320
- The and operators in (7.124) are not the
same as those appearing in (7.122) and earlier; they should appear as
and and denote transitions between and
. This text should appear after (7.124):
where and denote transitions between and , and we assume that higher order motional states are unoccupied.

- pp 370
- Equation (8.61) should have a sum over the index on
the right hand side.
- pp 380
- The right hand side of Equation (8.107) should read
``
''.
- pp 407
- On the first line there is a right parenthesis missing
from
.
- pp 417
- Equation (9.118) should have on the right
hand side, not .
- pp 418
- Equation (9.124) is missing a : it should read
.
- pp 455
- Just before Exercise 10.31, ``To establish condition
(b), that '' should be ``To establish condition (b),
that ''.
- pp 468
- In Figure 10.11 the eighth generator of the stabilizer
for the Shor code should be , not .
- pp 573
- Six lines after Equation (12.154), ``
'' should be ``
''.
- pp 588
- Item 7 in Figure 12.13, ``... will to serve as a
check...'' should omit the ``to''.
- pp 650
- Reference [BFC+96] should have page 2818, not 2828.
- pp 652
- Entry [BS98] should read
*IEEE Trans. Inf. Theory*instead of*itit*. - pp 667
- Index entry for ``element of reality'' should point to
page 112, not page 111.
- pp 670
- There should be an index entry ``majorization, 573''.

First Printing (September, 2000)

- Cover blurb
- ``quatum teleportation'' should be ``quantum
teleportation''.
- pp xvii
- (line 21) ``Apendix 2'' should be ``Appendix 2''.
- pp 112
- (9th line from bottom) ``co-authored with Nathan Rosen and
Boris Podolsky'' should be ``co-authored with Boris Podolsky and
Nathan Rosen''.
- pp 121
- (20 lines from bottom) ``[...] instinctively know what
problems, what techniques, and most importantly what problems are of
greatest interest to a computer scientist.'' should be ``[...]
instinctively know what techniques and, more importantly, what problems
are of greatest interest to a computer scientist.''
- pp 176
- Exercise 4.11 should be replaced by
Suppose and are non-parallel real unit vectors in three dimensions. Show that an arbitrary single qubit unitary may be written as

(4.13) for appropriate choices of and , .

- pp 168
- (15th line from bottom) The first reference
to Bennett's work on Maxwell's Demon should be amended. It currently
is to the paper [BBBW82], which is actually about quantum
cryptography. The correct reference is:
[Ben82a] C. H. Bennett, ``The thermodynamics of computation - A review'', Int. J. Theor. Phys.

**21**, 905-940 (1982). - pp 202
- In equation (4.87), should be .
- pp 202
- (5th last line of section 4.5.5) The text ``[...] this
means that quantum computers do not violate the Church-Turing thesis
that any algorithmic process can be simulated efficiently using a
Turing machine'' should not include the word ``efficiently''.
- pp 325
- In the second paragraph `` tesla'' should read
`` Tesla''.
- pp 424
- (5th line from the top) The term ``monotonicity of the
trace distance'' should be replaced by ``contractivity of the trace
distance''. Note that these two terms are often used interchangeably
in quantum computation and quantum information, however in our
presentation in the book we prefer the latter.
- pp 521
- Exercise 11.25 has a couple of typos in the first two
sentences. They should read as follows:
``We obtained strong subadditivity as a consequence of the concavity of the conditional entropy, . Show that the concavity of the conditional entropy may be deduced from strong subadditivity.''

- pp 570
- In Figure 12.9, there are missing boxes around
and , at the top and bottom of the figure, respectively.
- pp 621
- In Equation (A3.7) the two s on the left hand
side should be .
- pp 622
- In Equation (A3.16) there is a missing right parenthesis
just before the plus sign on the right hand side of the equation.
- pp 653
- The reference [DiV95a] should have the arXive number
removed, as it did not appear as a preprint at the archive.
- pp 663
- The reference [VR89] is incorrect. The given issue and page numbers are
for a different article, ``k-photon Jaynes-Cummings model with coherent
atomic preparation: Squeezing and coherence,'' by W. Vogel and D.-G.
Welsch. The correct citation is
K. Vogel and H. Risken, Phys. Rev. A

**40**(5): 2847-2849, 1989.

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