Error Correction of Quantum Reference Frame InformationError
Correction of Quantum Reference Frame Information
Patrick Hayden,1 Sepehr Nezami,1,2 Sandu Popescu,3 and Grant Salton
1,2,4,5,*
1 Stanford Institute for Theoretical Physics, Stanford University,
Stanford, California 94305, USA
2 Institute for Quantum Information and Matter, Caltech, Pasadena,
California 91125, USA
3 H. H. Wills Physics Laboratory, University of Bristol, Tyndall
Avenue, Bristol BS8 1TL, United Kingdom
4 Amazon Quantum Solutions Lab, Seattle, Washington 98170,
USA
5 AWS Center for Quantum Computing, Pasadena, California 91125,
USA
(Received 11 February 2019; revised 5 August 2020; accepted 12
January 2021; published 18 February 2021)
The existence of quantum error-correcting codes is one of the most
counterintuitive and potentially tech- nologically important
discoveries of quantum-information theory. In this paper, we study
a problem called “covariant quantum error correction”, in which the
encoding is required to be group covariant. This prob- lem is
intimately tied to fault-tolerant quantum computation and the
well-known Eastin-Knill theorem. We show that this problem is
equivalent to the problem of encoding reference-frame information.
In standard quantum error correction, one seeks to protect abstract
quantum information, i.e., information that is inde- pendent of the
physical incarnation of the systems used for storing the
information. There are, however, other forms of information that
are physical—one of the most ubiquitous being reference-frame
informa- tion. The basic question we seek to answer is whether or
not error correction of physical information is possible and, if
so, what limitations govern the process. The main challenge is that
the systems used for transmitting physical information, in addition
to any actions applied to them, must necessarily obey these
limitations. Encoding and decoding operations that obey a
restrictive set of limitations need not exist a priori.
Equivalently, there may not exist covariant quantum
error-correcting codes. Indeed, we prove a no- go theorem showing
that no finite-dimensional, group-covariant quantum codes exist for
Lie groups with an infinitesimal generator [e.g., U(1), SU(2), and
SO(3)]. We then explain how one can circumvent this no-go theorem
using infinite-dimensional codes, and we give an explicit example
of a covariant quantum error-correcting code using continuous
variables for the group U(1). Finally, we demonstrate that all
finite groups have finite-dimensional codes, giving both an
explicit construction and a randomized approximate construction
with exponentially better parameters. Our results imply that one
can, in principle, circumvent the Eastin-Knill theorem.
DOI: 10.1103/PRXQuantum.2.010326
I. INTRODUCTION
One of Shannon’s original insights in the formulation of
information theory was to focus on the transmission of sequences of
symbols, such as strings of 0’s and 1’s, with- out regard to the
semantic content of the message. This approach makes it possible to
encode an enormous variety of messages, from phone numbers to
photos, as long as the original information can be faithfully
represented in terms of a sequence of symbols. The same situation
exists in
*
[email protected]
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Fur- ther
distribution of this work must maintain attribution to the
author(s) and the published article’s title, journal citation, and
DOI.
the quantum world: quantum-information theorists are pri- marily
concerned with information that can be stored in a system of qubits
(or larger quantum systems), independent of the type of
information.
Here we study a situation in which the information is physical and
cannot be represented simply as abstract qubits. Consider the
following purely classical scenario [1]. Alice wishes to transmit
some directional information to Bob, e.g., the axis of rotation of
a gyroscope indicated by the vector n, so that Bob can prepare a
gyroscope rotat- ing around the same axis as Alice’s. If Alice and
Bob share a reference frame, Alice can measure different components
of n and describe the result in words to Bob, who then pre- pares
his own gyroscope to match. However, if Alice and Bob do not share
a reference frame, i.e., they do not know the relative alignment of
their coordinate systems, then this task is impossible. Without a
shared reference frame, Alice has no way to communicate a set of
symbols to Bob
2691-3399/21/2(1)/010326(17) 010326-1 Published by the American
Physical Society
indicating the axis of rotation of her gyroscope. Another simple
example is clock synchronization, wherein two distant observers
want to synchronize their clocks, but it is not possible to do so
by sending purely symbolic messages [2].
Of course, the simple examples described above do not mean that
sending physical information is impossible. For example, in the
classical example, Alice can prepare and send a physical copy of
her gyroscope to Bob, thereby indicating her direction. In this
way, Alice and Bob can even establish a shared reference frame.
Similarly, in the clock synchronization problem Alice can send a
copy of her clock to Bob [3] to establish a common time stan- dard
(ignoring relativistic effects). Quantum mechanically, Alice can
send direction information by sending polarized spins, while timing
information can be sent using quan- tum clocks such as two-level
atoms. As is common in the quantum case, many interesting and
counterintuitive effects occur. For example, sending two
antiparallel spins polarized along the desired direction is a
better direction indicator than sending two parallel spins [1,4].
The prob- lem of aligning quantum reference frames has garnered
significant attention in recent years [1,3–13].
In this paper we are interested in quantum error cor- rection of
physical information. Crucially, physical infor- mation can only be
communicated using systems that themselves have the physical
property of interest. This restriction constrains the actions that
can be performed on the physical systems, since we cannot, for
example, destroy or change physical information arbitrarily. In
par- ticular, there may be constraints on the set of possible
encoding and decoding schemes that one might have used to make the
system more robust to errors, thereby limit- ing our ability to
perform quantum error correction. In this paper, we characterize
the constraints placed on quantum error correction of physical
information.
In each of the examples described above, Bob’s lack of knowledge
about Alice’s reference frame or time stan- dard is modeled by the
action of an unknown element of some group on Alice’s state. For
directional reference frames, Alice and Bob are related by an
unknown rota- tion [i.e., an element of SO(3)], whereas in the
example of clock synchronization their clocks are related by an
unknown time translation [i.e., an element of U(1)]. In the spirit
of Ref. [14] (which generalizes reference-frame information to
general resource theory of asymmetry [14– 19]), we study error
correction of physical information that transforms under an
arbitrary group G. An impor- tant reduction following from the
analysis of Ref. [14] is that the existence of encoding schemes for
this type of information is equivalent to the existence of
ordinary, yet G-covariant, encoding schemes, which can correct the
same errors.
In this paper, we first study the case in which the group G has at
least one infinitesimal generator. In this first
case, we find a result strikingly different from conven- tional,
abstract quantum information: we prove a no-go theorem showing that
it is impossible to encode physi- cal information in any number of
finite-dimensional sys- tems such that the encoding allows for
perfect correction of any erasure error. We then show that both
condi- tions of the no-go theorem are necessary by construct- ing
codes that circumvent the theorem when either of the conditions is
violated. Specifically, we first demon- strate how one can encode
physical information to protect against erasure errors when one
uses continuous-variable modes (with infinite-dimensional Hilbert
spaces). Since continuous-variable modes are used, we expect this
result to be of practical interest. We then construct a per- fect
encoding scheme for any finite group G into finite- dimensional
spaces, which is again robust to erasure errors. Finally, we study
a family of group-covariant random codes and show that they can
provide encoding schemes with better parameters than the perfect
schemes for finite groups.
It is worth noting that the covariant channel formula- tion of the
problem is closely related to other results in the literature that
have very different motivations, includ- ing the Eastin-Knill
theorem [20] and recent studies of invariant perfect tensors [21].
We present a more detailed comparison in the discussion.
II. REFERENCE-FRAME ERROR CORRECTION
We begin with a description of error correction of spatial
reference frames, which corresponds to G = SO(3); the
generalization to other groups is immediate. Suppose Alice and Bob
share a (possibly noisy) quantum channel. Alice wants to
communicate some directional quantum informa- tion (a single spin,
say) to Bob, but Alice and Bob do not share a common reference
frame. Specifically, their reference frames are related by an
unknown rotation R ∈ SO(3). Alice and Bob will claim success if Bob
receives the spin in the same direction that it was sent by Alice
(i.e., the directional information is unchanged—a condition they
could check at a later stage). If the task is successful, Bob can
use the received spin to establish a shared reference frame, among
other things.
When sending quantum information through a noisy channel, the
information can be corrupted. This is also true of directional
information, so we wish to error correct this type of information.
We fix our error model to be erasure of a single spin (or mode),
and our goal is to design an error-correcting code to protect the
directional information from this noise.
To simplify the presentation, we focus on an encoding of one spin
into three (see Fig. 1) without loss of general- ity. We emphasize
that this choice of one into three is just for clarity—our results
hold for an arbitrary one-to-many
010326-2
encoding. We split the process into six steps.
1. Figure 1(a). Alice starts with an unknown input state ρin, a
density operator on Hin, representing some directional information.
Alice encodes this initial state using an encoding channel EA. We
use the sub- script A to indicate that EA is the encoding map in
Alice’s reference frame, and to distinguish it from the map as seen
in Bob’s frame: EB, to which we return shortly. Thus, the encoded
state σ123 on three spins is given by σ123 = EA(ρin).
2. Figure 1(b). Spin j ∈ {1, 2, 3} is lost. This is an era- sure
error of any one of the spins, but we assume that Bob can infer
which.
3. Figure 1(c). Prior to the erasure error, the encoded state as
seen by Bob would be U1 ⊗ U2 ⊗ U3σ123U†
1 ⊗ U† 2 ⊗ U†
3, where Ui = Ui(R) is a uni- tary representation of the unknown
rotation R mapping Alice’s coordinate system to Bob’s. Bob then
receives the state trj (U1 ⊗ U2 ⊗ U3σ123U†
1 ⊗ U†
2 ⊗ U† 3).
4. Figure 1(d). Bob decodes the state with an R- independent
decoding map Dj to obtain
Dj
[ trj (U1 ⊗ U2 ⊗ U3σ123U†
1 ⊗ U† 2 ⊗ U†
3) ] , in Bob’s
reference frame. If the protocol is successful, this state should
be equal to ρin = UinρinU†
in in order to match Alice’s original state, where Uin = Uin(R) is
the representation of the rotation group acting on the initial
state, and the tilde signals that this is the input state as seen
from Bob’s rotated reference frame.
5. Figure 1(e). Bob sends the decoded state through a hypothetical
perfect channel to Alice for verifica- tion.
6. Figure 1(f). Success is claimed if the received state is the
same as the initial state in Alice’s frame.
Using ρin = U† inρin Uin, the success condition becomes
ρin =Dj {
inρinUin)U † 1⊗U†
2⊗U† 3
(1)
for all R ∈ SO(3), states ρin ∈ Hin, and j ∈ {1, 2, 3}.
III. COVARIANT ERROR CORRECTION
Covariant quantum error correction is a seemingly dif- ferent
problem in which the encoding map is required to commute with the
action of the group. Continuing the example of mapping a single
spin into three, the covariance requirement is that the encoding
map satisfies
U1 ⊗ U2 ⊗ U3 E(U† inρin Uin)U
† 1 ⊗ U†
(e) (f)
FIG. 1. Setup: Alice wants to send a spin to Bob, but Alice and Bob
do not share a reference frame. (a) Alice encodes her spin into an
error-correcting code. (b) The environment erases one of the spins.
(c) Bob receives the encoded spins in his reference frame. (d) Bob
then decodes the remaining spins to reveal the original state. (e)
Bob sends the decoded spin using a (hypothet- ical) perfect channel
to Alice for verification. (f) Alice confirms that the recovered
state is the same as her original state.
for all R ∈ SO(3) and initial states ρin. In this version of the
problem Alice and Bob are assumed to share a single reference
frame. Imposing the simple constraint, Eq. (2), on the encoding
map, however, defines an error-correction problem equivalent to
reference-frame erasure correction, as we now see.
Let us return to the setting of reference-frame error cor- rection
momentarily. Alice performs the encoding EA in her reference frame.
In Bob’s reference frame, this opera- tion is denoted by EB,R (EB,R
is the quantum channel corre- sponding to the operation Alice
performs as seen in Bob’s reference frame). For a fixed EA in
Alice’s reference frame, EB in Bob’s frame is still parametrized by
the unknown rotation R, i.e., EB = EB,R. Specifically, EB,R(ρin) =
U1 ⊗ U2 ⊗ U3 EA(U†
inρin Uin) U† 1 ⊗ U†
2 ⊗ U† 3. The success
condition simplifies to
Dj {trj [EB,R(ρin)]} = ρin, (3)
for all states ρin and j ∈ {1, 2, 3}. Now introduce the average
channel E = ER[EB,R],
where the average is over all rotations R ∈ SO(3) accord- ing to
the Haar measure. By the linearity, the error- correction relation
Eq. (3), holds for the average channel: Dj {trj [E(ρin)]} = ρin.
Moreover, the averaged channel is clearly covariant in the sense of
Eq. (2), provided we substitute ρin for ρin in the equation.
Thus if reference-frame error correction, Eq. (1), is possible, we
find a covariant erasure-correcting encod- ing. Moreover, it is
straightforward to confirm that by choosing E to be EA, Eqs. (3)
and (2) lead to Eq. (1). There- fore, reference-frame error
correction and covariant error correction are equivalent.
010326-3
HAYDEN, NEZAMI, POPESCU, and SALTON PRX QUANTUM 2, 010326
(2021)
IV. RESULTS
We now study a more general problem. Consider an encoding map E ,
which encodes an initial state on Hin into n encoded systems on
Hout = H1 ⊗ · · · ⊗ Hn. The output Hilbert spaces are arbitrary at
this point (i.e., they can be the same or different, finite or
infinite dimensional, etc.). Suppose there exists a group G, and
representa- tions Uin, U1, . . . , Un acting on the different
Hilbert spaces. Moreover, suppose that the channel is covariant
under the action of the group:
E(ρin) = U1 ⊗ · · · ⊗ UnE(U† inρinUin)U
† 1 ⊗ · · · ⊗ U†
n. (4)
Our goal is to answer the following question: is it possible to
recover the original state after erasure of an arbitrary set of at
most k subsystems (which we henceforth refer to as modes)?
We study this question in different scenarios.
1. G is a Lie group and the code is finite dimensional. We prove a
no-go theorem: no perfect covariant error-correcting scheme can be
implemented in this case. This applies to the example of sending
spins, as in the original reference-frame error-correction task. In
fact, the no-go theorem applies to all groups with at least one
infinitesimal generator, and it states that such generators can
only act trivially on encoded states.
2. G is a Lie group and the code is infinite dimen- sional. We show
that G-covariant error-correcting codes are possible when the
encoding uses infinite- dimensional systems. This illustrates the
existence of interesting error-correcting codes for a Lie group
when the conditions of the no-go theorem above are not satisfied.
We provide an explicit code for G = U(1) in Appendix A.
3. G is a finite group and the code is finite dimen- sional. For
any finite group G, we find examples of perfect covariant
error-correcting schemes. This is again consistent with our no-go
theorem since finite groups do not have infinitesimal generators.
We also provide a randomized construction in Appendix B to obtain
approximate codes with better parameters.
V. CASE 1: G IS A LIE GROUP AND THE CODE IS FINITE
DIMENSIONAL
Suppose that the local Hilbert space dimensions are all finite, and
that the group G is a Lie group [22]. Choose one infinitesimal
generator of the Lie group, without loss of generality. We denote
this generator acting on the input mode by Tin and on the ith
output mode by Ti. The gen- erator acting on the full set of output
modes is Tout = T1 + · · · Tn. Assume that Tin is nontrivial; our
goal is to
show that covariant quantum error correction is impossible with
this assumption.
Consider an initial state ρin and a slightly rotated state ρin(ε) =
e−iεTinρineiεTin . These states are encoded as σout = E(ρin) and
σout(ε) = E[ρin(ε)]. Using the fact that E(ρin)
is invertible on its range, we can find a set of orthogonal
isometries {Ei}, (E†
i Ej = δij I ) and probabilities pi such that
E(ρin) = ∑
i
piEiρinE† i
(see, e.g., [23], Theorem 10.1 and the proof using Hin as the code
space). The inverse channel E−1(σout) can be described by the same
set of isometries on the range of E
E−1(ρout) = ∑
† i . A crucial but elementary
property of E−1 is that if σout = E(ρin) and A is some arbitrary
operator, then E−1(Aσout) = E†(A)ρin, where E†(A) = ∑
i piE † i AEi. Expanding the relation ρin −
ρin(ε) = E−1[σout − σout(ε)] to first order in ε we obtain
[Tin, ρin] = E−1([Tout, σout])
= [E†(Tout), ρin]. (5)
Assuming error correction succeeds, we can recover the original
state from any of the n − k subsets of the encoded modes. In other
words, upon tracing out all output modes except the ith mode, the
reduced state ρi must be indepen- dent of the initial state.
Thus, for any state ρin, we find that tr(Tiσout) = αi, where αi is
independent of ρin. It is easy to see that
αi = tr(Tiσout) = tr[TiE(ρin)] = tr[E†(Ti)ρin]
for all ρin. Hence E†(Ti) ∝ I , and consequently E†(Tout) ∝ I .
This implies that the last term in Eq. (5) is zero, which means
that [Tin, ρin] = 0 for all ρin. In order for Tin to com- mute with
all ρin it must be trivial, which is a contradiction of our
assumption. We conclude that perfect recoverability is
impossible.
VI. CASE 2: G IS A LIE GROUP AND THE CODE IS INFINITE
DIMENSIONAL
If we allow Alice to use infinite-dimensional Hilbert spaces
(violating one of the hypotheses of our no-go theorem), then even a
naïve solution to the problem exists. Intuitively, a simple way to
achieve the task is for Alice to append a classical gyroscope to
the encoded state that she sends to Bob [24]. Bob can then infer
information about Alice’s reference frame by measuring the state
of
010326-4
COVARIANT QUANTUM ERROR CORRECTION... PRX QUANTUM 2, 010326
(2021)
the gyroscope, thereby establishing a common reference frame.
Indeed, this is one strategy we outline below. Since the full state
is sent through the noisy channel, Alice must actually send two
gyroscopes in order to safeguard against loss of one of the encoded
shares [25].
In the reference-frame error-correction paradigm, Alice chooses her
favorite (noncovariant) erasure code, encodes, and then appends two
redundant ancilla (the classical gyro- scopes) indicating her
reference frame to the encoded state. The ancilla must necessarily
be states in infinite- dimensional Hilbert spaces so that the no-go
theorem does not apply (and in this protocol this is also necessary
so that Alice can specify her reference frame with perfect preci-
sion) [26]. If any shares of the erasure code are lost, Bob can
first measure the gyroscopes to learn Alice’s reference frame, and
then use the standard decoding on the remaining shares in the
aligned frame. Since Alice sent two ancilla, one can freely be lost
without failure.
Let us now study this problem in the covariant quantum
error-correction paradigm. Let HG = span{|g}, where g ∈ G, and the
set {|g} forms an orthonormal basis for HG. The group acts via U(g)
|h = |gh [27]. To encode her state, Alice chooses her favorite,
noncovariant erasure- correcting code (denoted by E0) [e.g., the
C
3 → (C3)⊗3
qutrit code] without loss of generality. As before, define the
rotated encoding map (i.e., the map in Bob’s frame) by
Eg() = U(g)⊗3E0 [ U†(g)U(g)
] U†⊗3(g). (6)
To complete the encoding, Alice appends two ancilla in the state |e
e| (where e ∈ G is the identity element) for a full encoded state
E0() ⊗ |e e|⊗2 as seen in her frame. The two |e e| registers
represent the classical gyroscopes above. The encoding is made
covariant by averaging over the group G. Thus, the full encoding is
defined by symmetrizing the channel and ancilla together:
E() = ∫
g∈G dg Eg() ⊗ |g g|⊗2,
which is clearly covariant. Decoding is then fairly simple: one
need only measure
any surviving ancilla, which collapses the state to one cor-
responding to the measured group element. Bob can then recover the
encoded state from the surviving shares of the code.
The procedure described above is not the only method one can use in
this case. In Appendix A we describe an explicit, group-covariant,
continuous-variable quantum erasure code for the example of G =
U(1). An input con- tinuous variable mode is mapped into three
physical modes via the encoding
EU(1) = ∑ x,y∈Z
|−3y, −x + y, 2(y + x)123 x|in.
We leave all relevant details to Appendix A.
VII. CASE 3: G IS A FINITE GROUP AND THE CODE IS FINITE
DIMENSIONAL
Consider a finite group G. Here we show that there exist
G-covariant channels that encode the input Hilbert space into
finite-dimensional Hilbert spaces, while satisfying the
erasure-correction conditions.
Suppose the group G acts on some set A. By definition, the action
of G permutes the elements of A. Our goal is to construct an
error-correction scheme for which the action of the group commutes
with the process of encoding, era- sure, and decoding. To achieve
our goal, we first start with a noncovariant code. We then consider
a tensor product of many copies of this noncovariant code, one
tensor factor for each element of A. This new code is already a
covari- ant code! To see this, note that the encoding acts as a
tensor product over the factors, while the group action simply
permutes the factors. Therefore, the encoding map and the group
action commute, which implies that the encoding is G
covariant.
To be more precise, consider a channel E0 : S(Hin) → S(Hout :=
H⊗n), where S(H) denotes the space of den- sity matrices on the
Hilbert space H. Suppose that E0 is an encoding map that allows for
recovery after erasure of an arbitrary set of k of the n output
modes. We make no assumptions about the covariance of E0—it is an
arbitrary erasure-correcting map. We now introduce a new
encoding
E = ⊗
0 , E : S(H⊗|A| in ) → S(H⊗|A|
out ),
where we use ⊗
a∈AE0 to indicate that the different tensor copies are labeled by
elements of A. For each g ∈ G the action of the representation on
H⊗|A| is defined by
U(g) |φa1 |φa2 · · · |φa|A| = |φg−1a1 |φg−1a2
· · · |φg−1a|A| .
Here a1 · · · a|A| is a list of the elements of A. The covariance
of E follows from the definition, and the error-correction
properties of E are directly inherited from those of E0. Therefore,
we succeed in finding a perfect G-covariant channel. Figure 2 shows
an example in which G = S3 (the permutation group on three
elements) and A = {1, 2, 3}.
While our construction can be formally extended to infinite groups
with their associated infinite-dimensional representations, we have
not determined which additional conditions need to be imposed in
order for the argument to remain mathematically rigorous.
The construction presented in this section provides codes in which
the Hilbert spaces can be exponentially large in |G|. However, it
is known that in many cases random codes give near optimal
error-correcting schemes with good parameters [28–32]. In Appendix
B, we show
010326-5
HAYDEN, NEZAMI, POPESCU, and SALTON PRX QUANTUM 2, 010326
(2021)
FIG. 2. Permutation covariance for the group S3 acting on S3 (i.e.,
G = A = S3). Each fork represents a code that maps one qudit into
three, and can correct an erasure error on any one output qudit.
π12 ∈ G is the transposition that swaps sys- tems 1 and 2. Left:
the map E[Uin(π12)ρinUin(π12)
†]. The group action permutes the inputs to the channel. Right: the
map Uout(π12)E(ρin)Uout(π12)
†. As it is evident from the wiring of the forks, these two maps
are equivalent.
that choosing a random, covariant isometry yields approx- imate
error-correcting codes for which the dimension of each mode is |G|.
For these codes, the worst-case fidelity of recovery, Fworst,
behaves well with high probability. Specifically, P(Fworst < 1 −
ε) decays exponentially in |G|. For example, we show in the
Appendix E
P (
) ≤
)]} .
(7)
It is clear that for n ≥ 5 and |G| sufficiently large, the expo-
nent on the right-hand side becomes arbitrarily negative,
indicating that the worst-case fidelity of recovery is close to 1
with high probability.
VIII. DISCUSSION
We showed that perfect error correction of physical information
against erasure is a process that depends on the details of the
symmetry group and dimensions of the code. For example, covariant
quantum error correction is impos- sible when the symmetry group is
a Lie group and the code is finite dimensional. This is connected
to the following no-go theorems in the literature:
(a) Eastin-Knill theorem [33]. Eastin and Knill proved [20] that it
is not possible to encode information in an error-detecting code
such that a set of uni- versal gates can be implemented
transversally. We can reproduce the main thrust of the Eastin-Knill
theorem [34] using an instance of our no-go theorem in which the
input space is the set of N logi- cal qudits, the output consists
of physical qudits, and letting the group be G = U(N ). Moreover,
our continuous-variable code construction provides a demonstration
that the Eastin-Knill theorem can
be circumvented in principle, although our explicit examples do not
immediately appear to be useful for fault-tolerant quantum
computation.
(b) Invariant perfect tensors. A quantum state on the tensor
product of a number of Hilbert spaces is a perfect tensor if, for
any bipartition of the Hilbert space into two collections of
constituent factors, it forms an isometry from the smaller space to
the larger [35]. Motivated by the construction of phys- ical states
in the Hilbert space of loop quantum gravity, the authors in Ref.
[21] defined the notion of invariant perfect tensors as those
perfect tensors that are invariant with respect to the action of
SU(2). The authors proved that there are no invariant perfect
tensors with four tensor factors. This can be seen as a direct
consequence of our no-go theorem for G = SU(2), by considering a
four-partite invariant perfect tensor as a one-mode to three-mode
isome- try. Such an invariant perfect tensor with four tensor
factors would define an SU(2)-covariant erasure- correcting code,
which is prohibited by our no-go theorem. Furthermore, our no-go
theorem states that there are no invariant perfect tensors with
higher numbers of tensor factors, thereby solving an open question
in Ref. [21].
One might hope to find a more quantitative relation between some
measure of the size of the group and the dimension of the code when
error correction is possible. For example, a condition of the form
|G| ≤ dim(code) (i.e., dimension of the physical Hilbert space) is
consis- tent with our no-go theorem and the examples in cases 1 and
2.
ACKNOWLEDGMENTS
We thank Dawei Ding, Iman Marvian, Michael Walter, and Beni Yoshida
for helpful discussions. S.N. acknowl- edges support from Stanford
Graduate Fellowship. G.S. acknowledges support from a NSERC
postgraduate schol- arship. This work is supported by the CIFAR and
the Simons Foundation. The work of G.S. was performed before
joining Amazon Web Services.
Note added.—Recently, there has been follow-up work on covariant
quantum error correction (e.g., Refs. [36– 40]). In Ref. [36],
approximate versions of the no-go theorem above and of the
Eastin-Knill theorem were developed, in which the recovery fidelity
is bounded above by a function of the symmetry group G and the code
dimension. These approximate theorems address our earlier questions
about a quantitative relation between group size and code
dimension. Moreover, several of the papers listed above attempt to
address the question of practical fault-tolerant quantum
computation, either using
010326-6
approximate quantum error correction, or approximate
computation.
APPENDIX A: G = U(1) AND THE CODE IS CONTINUOUS VARIABLE
Here we provide an explicit U(1)-covariant 1 → 3 encoding. The
construction presented in this section does not violate the no-go
theorem stated in case 1 above as the local systems are infinite
dimensional. Since the symmetry group in question is U(1), this
code could be implemented in optical modes, and it is arguably more
natural than the construction presented in case 2.
We take the Hilbert space to be the space of functions on a circle
using the position basis {|φ}φ∈[0,2π). U(1) acts on this space via
the regular representation: if g = eiθ ∈ U(1), then the action of
the regular representation is defined by U(g) |α = |α + θ. It is
convenient to work in the Fourier basis where the Hilbert space is
described by the con- jugate momentum basis {|n}n∈Z and the group
acts by U(g) |n = einθ |n. We define the isometry to be the fol-
lowing operator expressed in the conjugate momentum basis
EU(1) = ∑ x,y∈Z
More explicitly, the isometry maps the state ∑
x φ(x) |xin to |123 = ∑
x,y φ(x) |−3y, −x + y, 2(y + x)123. It is easy to see that this
isometry is U(1) covariant:
U(g)⊗3EU(1)U(g)† = ei[−3y−x+y+2(y+x)]EU(1)e−ix
= EU(1).
Here we show, step by step, that this mapping can correct an
erasure error. Consider the encoded density matrix
123 = ∑
φ(x1)φ(x2) ∗ |−3y1, −x1 + y1, 2(y1 + x1)
−3y2, −x2 + y2, 2(y2 + x2)|123 .
We study the loss of modes 1, 2, and 3, in turn.
1. Loss of the first mode. The resulting density matrix is
23 = ∑
φ(x1)φ(x2) ∗ |−x1 + y, 2(y + x1)
−x2 + y, 2(y + x2)|23 .
Decoding starts with the linear map |a, b → |a, b − 2a,
yielding
∑ x1,x2,y∈Z
φ(x1)φ(x2) ∗ |x1 + y, 4x1 x2 + y, 4x2|23.
We then use an isometry, which maps the states of the form |a, 4b
to |a, b
∑ x1,x2,y∈Z
Finally, by |a, b → |a − b, b, we obtain
∑ x1,x2,y∈Z
Therefore, tracing out mode 2 reveals the original state.
2. Loss of the second mode. The resulting density matrix is
13 = ∑
or, equivalently by the change of variable y → y + x1,
13 = ∑
φ(x1)φ(x2) ∗ |−3(y + x1), 2(y + 2x1)
−3(y + x2), 2(y + 2x2)|13 .
We now use an isometry, which maps states of the form |3a, 2b to
|a, b
∑ x1,y,x2∈Z
By |a, b → |a, 2a + b, we have
∑ x1,y,x2∈Z
We now use |a, b → |−(a + b), b to obtain
∑ x1,y,x2∈Z
Tracing out mode 3 reveals the original state.
010326-7
HAYDEN, NEZAMI, POPESCU, and SALTON PRX QUANTUM 2, 010326
(2021)
3. Loss of the third mode. Again, the resulting density matrix
is
12 = ∑
Using the change of variable y → y + x1 we have
12 = ∑
φ(x1)φ(x2) ∗ |−3(y − x1), −2x1 + y
−3(y − x2), −2x2 + y|12 .
Applying an isometry that maps |3a, b to |a, b yields
∑ x1,y,x2∈Z
Using |a, b → |a, a + b, ∑
x1,y,x2∈Z
|−(y − x1), −x1 −(y − x2), −x2|12 .
Finally, the isometry |a, b → |a + b, −a turns the state to
∑ x1,y,x2∈Z
Thus we can recover the state on mode 2.
APPENDIX B: G IS A FINITE GROUP AND THE CODE IS A RANDOM
G-COVARIANT ISOMETRY
In the construction presented for case 3, the local Hilbert space
dimension can grow exponentially with |G|. In this section we
present an alternative, approximate method for error correction in
which the local Hilbert space dimen- sions are equal to |G|. Our
goal is to prove Eq. (7) of the main text.
Consider a 1 → n encoding. We look for isometries that map HG →
H⊗n
G , where HG denotes the Hilbert space associated to the regular
representation of G with the basis {|g}g∈G. Thus dimHG = |G| = d.
We represent the action of the regular representation of g ∈ G on
HG by U(g).
To construct a random covariant map, we start with a random
invariant state | ∈ H⊗(n+1)
G . For our purposes, a random state is one that is chosen randomly
with respect to the unitary invariant measure; random unitaries are
uni- taries chosen randomly with respect to the Haar measure;
and a state is invariant if U(g)⊗(n+1) | = | for all g ∈ G. By
projecting our chosen state onto an un-normalized, maximally
entangled state |φ+AB = ∑ |iA |iB we obtain a map E (which is close
to an isometry with high probabil- ity) from Hin → H⊗n,
Ein,1···n = √
d φ+|in,0 |0···n .
Note that the covariance of E defined by U(g)⊗nE = EU(g), which
follows from the invariance of |. From E we can define the exact
isometry T as
T := E(E†E)−1/2.
One can verify that T is also a covariant map, since [E†E, U(g)]
for all g ∈ G. Our encoding is then defined by
E(ρin) = TρinT†.
With the covariant encoding in hand, we now turn our attention to
the decoding. Before diving in, let us first define two notational
conventions that are used frequently henceforth. Firstly, we use
trx to indicate tracing out all subsystems except the set x.
Secondly, if there are two isomorphic Hilbert spaces Hα and Hβ with
the same pre- ferred basis, and if the operator Xα acts on Hα ,
then by (Xα)β we mean the operator Xα acting on Hβ (in the sense
that the matrix corresponding to Xα is simply applied to Hβ). One
can think of (Xα)β as overriding the Hilbert space indices. When it
is clear to do so, we use Xβ instead of (Xα)β for brevity.
To decode after loss of one of the modes, say mode 1 without loss
of generality, Bob first replaces the lost mode by a maximally
mixed state τ1 and then decodes the state τ1 ⊗ tr1 [E(ρin)]. The
decoding map is given by
σout = D1(ρ12···n)
23V23···n)ρ12···n(UT 23V23···n)†]}out,
where U01, and V23···n are unitaries that transform |0···n into its
Schmidt form:
U01 ⊗ V2···n |0···n = ∑
i,j
√ λij |ij 01 ⊗ |ij 0 · · · 023···n ,
and U23 = (U01)23 is the same operator as U01 but acting on the
Hilbert spaces indexed by 2 and 3. In other words, U01 =
(U23)01.
With the decoding above, our task is now to prove Eq. (7) of the
main text. Our first step is bounding the worst- case fidelity of
recovery Fworst in terms of the distance between 01 (the reduced
density matrix of the invariant state |) and the maximally mixed
state.
Lemma 1. For 0 ≤ ε ≤ 1, if 01 − τ01∞ ≤ (ε/3d2), then 1 − ε ≤
Fworst.
010326-8
Proof. We prove this in three steps.
(a) Step 1. We first simplify the expression for the recovered
state and show that
D1[τ1 ⊗ E(ρin)]
.
(b) Step 2. We then use joint concavity of the fidelity, and
properties of the Schatten norm to bound the
worst-case fidelity
.
From the above equation, it is already clear that if 0 and 01 are
close to the maximally mixed state, then the worst-case fidelity
will be close to 1. We quantify this in the last step.
D1[τ1 ⊗ E(ρin)] = tr2
23
] . (C1)
Using the fact that E†E = d(T 0 )in, and the definition ρin =
1/d
( T
) in
, we have that TρinT† = EρinE†. From the definition of E we can
simplify the formula for the encoding map:
E(ρin) = EρinE† = d tr0 (| |0···n ρT
0
( UT
23ρ T 0
However, recall that U01 ⊗ V2···n |0···n = ∑ i,j
√ λij |ij 01 ⊗ |ij 0 · · · 023···n, and that U01
1/2 01 U†
V2···n |0···n = 1/2 01 U†
01 |φ+02 |φ+13 |0 · · · 04···n = U∗ 23
( T
D1[τ1 ⊗ E(ρin)]
(
APPENDIX D: STEP 2
Our goal now is to lower bound the fidelity of recovery. Since the
fidelity is jointly concave, we know that the minimum fidelity of
recovery for the channel is achieved with a pure input state, say
ρ0 = (|κ κ|)T, where we add the transpose to simplify the
expressions. In this case, the recovered state takes the following
form:
010326-9
HAYDEN, NEZAMI, POPESCU, and SALTON PRX QUANTUM 2, 010326
(2021)
D1[τ1 ⊗ E(ρin)] = tr1
1/2 01
Fmin = min |κ
√ tr ( κ|0
−1/2 0
) .
To proceed, we use the following basic property of the Schatten
norm: for (1/p) + (1/q) = 1, Yp ≥ | tr(XY†)| if X q = 1. Applying
this inequality when X = I1/
√ d and p = q = 2 we find
κ| 1/2 01
−1/2 0 |κ
This concludes step 2.
APPENDIX E: STEP 3
We ultimately want to lower bound the worst-case fidelity using
concentration of measure techniques for 01 and 0. We start by upper
bounding
tr1
(
(
∞
,
where |g, g ∈ G form a basis for evaluating the trace, the first
inequality is the triangle inequality, and the second inequality
comes from the fact that the infinite Schatten norm of a Hermitian
operator is equal to its maximum eigen- value. Now, one can check
that for any λ ≥ 0, |λ1/2 − 1/d| ≤ d|λ − 1/d2|. Taking {λi} to be
the set of eigenvalues of
1/2 01 , and using the aforementioned inequality, we obtain
1/2 01 − I01
∞
:one can simply check that for any real number λ such that |λ −
1/d| ≤ 1/2d, then
λ−1/2/ √
d − 1 ≤ d |λ − 1/d|. In particular, since this inequality holds for
all of the eigenvalues of 0,
(
COVARIANT QUANTUM ERROR CORRECTION... PRX QUANTUM 2, 010326
(2021)
(
,
(
−I0∞ ,
where the second inequality follows from the fact that XY∞ ≤ X ∞Y∞
for any pair of matrices X and Y. Using Eqs. (E1) and (E2)
above,
κ|0 tr1
− ( d2 01 − τ01∞
) (d 0 − τ0∞) .
Note that the condition 0 − τ0∞ ≤ (1/2d) is satisfied, since 0 −
τ0∞ ≤ d 01 − τ01∞ and 01 − τ01∞ ≤ (ε/3d2). Finally, since 0 − τ0∞ ≤
d 01 − τ01∞, we have that
κ|0 tr1
)2 ,
Fworst ≥ κ|0 tr1
)2 ≥ 1 − ε,
which proves the lemma. To complete the proof, it remains to be
shown that our assumption is valid. Specifically, in order to show
that the
worst-case fidelity is close to 1, it suffices to prove that the
reduced density matrix of random invariant states, 01, is very
close to the maximally mixed state in operator norm (i.e., 01 −
τ01∞ is small) with high probability. Since
01 − τ01∞ = max σ01
|tr [σ01(01 − τ01)]|,
where the maximization is done over all possible density matrices σ
, we can instead study the quantity on the right-hand side. To show
that this is small, we follow the techniques used in Refs.
[41–44].
Before stating the proof in its full glory, let us first gain an
imprecise, high-level overview of the strategy. We first define an
ε-net on the set of density matrices on H0 ⊗ H1, i.e., a finite set
of density matrices σ01 such that any other density matrix σ01 is
close to one of the elements of the net in the trace norm. If we
can then show that |tr [σ01(01 − τ01)]| is small for every σ in the
net, then it must be small for all density matrices σ01. Using
large deviation methods, we then
010326-11
HAYDEN, NEZAMI, POPESCU, and SALTON PRX QUANTUM 2, 010326
(2021)
prove that for any fixed density matrix σ01 (including the elements
of the net), |tr [σ01(01 − τ01)]| is small with very high
probability. Since the number of elements in the net is finite
(with a known upper bound), we can then use a union bound to show
that |tr [σ01(01 − τ01)]| is small for all elements in the net with
high probability. Therefore, we can bound |tr [σ01(01 − τ01)]|,
arriving at our desired conclusion.
We now give a detailed proof of Eq. (7) of the main text. Let
Pδ,σ01 be the probability that, for a fixed σ01, |tr [σ01(01 −
τ01)]| ≥ δ/d2, and let Pδ = maxσ01 Pδ,σ01 , where the maximum is
over all density matrices on H0 ⊗ H1. The following lemma relates
P
[01 − τ01∞ ≤ (ε/3d2) ]
P ( 01 − τ01∞ ≤ ε
.
Proof. Consider an α/3d2-trace distance net M of pure states in H0
⊗ H1, with α ≤ ε. For every pure state σ01, there exists a pure
state σ01 such that
σ01 − σ011 ≤ α
3d2 , (E3)
by definition. It is known that we can choose M such that |M| ≤ (
15d2/α
)2d2 [43, Lemma II.4]. Now if
|tr[σ01(01 − τ01)]| ≤ (ε − α/3d2), then from Eq. (E3) it follows
that
tr[σ01(01 − τ01)] ≤
tr[σ01(01 − τ01)] +
≤ ε − α
≤ ε − α
3d2
3d2
3d2
} . (E4)
P { ∃σ01 ∈ M :
3d2
≤ P ε−α 3
|M|.
This, along with Eq. (E4), conclude the proof of the lemma.
010326-12
COVARIANT QUANTUM ERROR CORRECTION... PRX QUANTUM 2, 010326
(2021)
In the next section, we use large deviation techniques to show
that
Pδ ≤ exp (−dn−2δ2/6
) for 0 ≤ δ ≤ 1. (E5)
We defer the proof to the next section, but we use the result
immediately. Combining Lemma 1, Lemma 2, and Eq. (E5) we have
P (Fworst ≤ 1 − ε) ≤ P ( 01 − τ01∞ ≥ ε
3d2
) ≤ min
3 × [
15d2
α
]2d2
( 15d2
α
)] .
One convenient choice of ε and α is ε = d(9−2n/8) and α = ε/2. With
this choice we find
P (Fworst ≥ 1 − ε) ≤ exp { − d2
216
[ d
( 30d
)]} ,
which reduces to Eq. (7) of the main text after substituting |G|
for d.
APPENDIX F: PROOF OF EQ. (E5)
The goal of this appendix is to prove Eq. (E5). The discussion is
split into two parts: we first explain the random invariant state
construction, and then we prove the desired bound.
APPENDIX G: CONSTRUCTION OF RANDOM INVARIANT STATES
Consider the invariant subspace of H⊗(n+1)—it is easy to see that
the invariant subspace is spanned by states of the form
1√ d
∑ g∈G
|gh1, gh2, . . . , ghn, g0···n.
We now introduce an isometry M from H⊗n to the invariant subspace
of H⊗(n+1),
M = 1√ d
∑ g,h1,...,hn
|gh1, gh2, . . . , ghn, g0···n h1, h2, . . . , hn|0···n−1.
The projector onto the invariant subspace is defined as 0···n = MM
†. 0···n has the important property that, upon tracing out any one
of the subsystems, it becomes the identity operator on the
remaining subsystems. That is
tri 0···n = I0···i···n. (G1)
A random invariant state |0···n is constructed by choosing a random
state |φ0···n−1 in H⊗n from the unitary invariant measure, and then
mapping |φ to H⊗(n+1) using the isometry M , |0···n = M
|φ0···n−1.
APPENDIX H: PROOF OF EQ. (E4)
To begin, we upper bound the moment generating function, E exp [t
tr (σ0101)], for an arbitrary density matrix σ01, where 01 = tr01
(0···n) and the average is over random invariant states |0···n.
Note that tr (σ0101) = tr (σ010···n) =
010326-13
HAYDEN, NEZAMI, POPESCU, and SALTON PRX QUANTUM 2, 010326
(2021)
tr ( σ01Mφ0···n−1M †
) . One can easily check that M †σ01M = σ G
01 ⊗ I2···n−1, where
σ G 01 = 1
2 h′ 1, h′
2| .
One can also check that σ G 01 is a density matrix, specifically a
version of σ01 symmetrized by the group G. Therefore,
tr (σ010···n) = φ| σ G 01 |φ ,
where |φ = |φ0···n−1 is a state on H⊗n chosen from the unitary
invariant measure (see the first subsection of this
appendix).
We now choose a Gaussian state |g0···n−1 in which the coefficients
of the wave function are chosen independent and identically
distributed from a complex Gaussian distribution centered at zero
with variance d−n. Thus E|gg2
2 = 1. Therefore, we have
E|g exp ( t g| σ G
01 |g ) = E|φEg2 exp ( t g2
2 φ| σ G 01 |φ )
≥ E|φ exp [ t ( Eg2g2
2
]
01 |φ)
= E exp [t tr (σ010···n)] ,
where the inequality follows from the convexity of the exponential
function. Now suppose that the eigenvalues of σ G
01 are pi0,i1 . Since the Gaussian states are unitarily invariant,
we can evaluate E|g exp
( t g| σ G
01 |g) in a basis in which σ G 01 ⊗ I2···n−1 is diagonal. In that
basis,
E|g exp ( t g| σ G
01 |g) = E|g exp
t
∏ i0···in−1
) .
However, the radial probability density for each coefficient is p
(|gi0···in−1 |) = 2dn|gi0···in−1 | exp (−dn|gi0···in−1 |2
) . Using the
Egi0···in−1 exp
) = 1
E|g exp ( t g| σ G
01 |g) = ∏ i0,i1
( 1 − t pi0,i1
.
Ultimately, we fix the value of t to prove the bound in Eq. (E4),
but we need to distinguish the cases in which t is positive or
negative to bound the fluctuations of tr (σ0101) above or below
1/d2. Therefore, we discuss these two different ranges for t
separately.
1. Positive t: We use the assumption that t is positive to limit
the fluctuations of tr (σ0101) above 1/d2. Let 0 < s < 1 be a
fixed number, and restrict t to 0 ≤ t ≤ sdn. Under these
conditions, we have,
( 1 − t pi0,i1
Combining with Eq. (H1), we have
E|g exp ( t g| σ G
01 |g ) ≤ E|g exp ( t g| σ G
01 |g) = ∏ i0,i1
( 1 − pi0,i1 t
P [
d2
] = P
( t 1 + δ
)] ,
where we use Markov’s inequality for the exponentials and Eq. (H1).
To obtain the best result, we now set t = sdn
and s = 1 − (1 + δ)−1/2. With this substitution,
P [
d2
] ≤ exp
,
where the last inequality is valid for 0 ≤ δ ≤ 1. 2. Negative
t:
We now use the constraint on t to limit the fluctuations of tr
(σ0101) below 1/d2. Assuming that s > 0 and −sdn ≤ t ≤ 0, one
can show that
( 1 − tpi0,i1
01 |g) = ∏ i0,i1
( 1 − pi0,i1 t
d2
] = P
( 1 d2 − δ
] ≥ exp (
]
]} .
We now fix t = −sdn and s = δ/(1 − δ) to get
P [
d2
] ≤ exp
}
010326-15
HAYDEN, NEZAMI, POPESCU, and SALTON PRX QUANTUM 2, 010326
(2021)
[1] N. Gisin and S. Popescu, Spin Flips and Quantum Infor- mation
for Antiparallel Spins, Phys. Rev. Lett. 83, 432 (1999).
[2] If the transmission time for each message is predeter- mined,
that provides a resource that could itself be used for clock
synchronization. To avoid this loophole, symbolic messages should
not be received at predetermined times.
[3] J. Preskill, Quantum clock synchronization and quantum error
correction, arXiv preprint quant-ph/0010098 (2000).
[4] S. Massar and S. Popescu, Optimal Extraction of Informa- tion
from Finite Quantum Ensembles, Phys. Rev. Lett. 74, 1259
(1995).
[5] E. Bagan, M. Baig, A. Brey, R. Munoz-Tapia, and R. Tar- rach,
Optimal encoding and decoding of a spin direction, Phys. Rev. A 63,
052309 (2001).
[6] R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams,
Quantum Clock Synchronization Based on Shared Prior Entanglement,
Phys. Rev. Lett. 85, 2010 (2000).
[7] C. Souza, C. Borges, A. Khoury, J. Huguenin, L. Aolita, and S.
Walborn, Quantum key distribution without a shared reference frame,
Phys. Rev. A 77, 032345 (2008).
[8] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner,
Degradation of a quantum reference frame, New J. Phys. 8, 58
(2006).
[9] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Classical and
Quantum Communication Without a Shared Reference Frame, Phys. Rev.
Lett. 91, 027901 (2003).
[10] A. Peres and P. F. Scudo, Entangled Quantum States as
Direction Indicators, Phys. Rev. Lett. 86, 4160 (2001).
[11] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reference
frames, superselection rules, and quantum information, Rev. Mod.
Phys. 79, 555 (2007).
[12] G. Gour and R. W. Spekkens, The resource theory of quan- tum
reference frames: Manipulations and monotones, New J. Phys. 10,
033023 (2008).
[13] I. Marvian and R. W. Spekkens, Modes of asymmetry: The
application of harmonic analysis to symmetric quantum dynamics and
quantum reference frames, Phys. Rev. A 90, 062110 (2014).
[14] I. Marvian and R. W. Spekkens, The theory of manipula- tions
of pure state asymmetry: I. basic tools, equivalence classes and
single copy transformations, New J. Phys. 15, 033001 (2013).
[15] I. Marvian and R. W. Spekkens, Asymmetry properties of pure
quantum states, Phys. Rev. A 90, 014102 (2014).
[16] I. Marvian and R. W. Spekkens, Extending noether’s theorem by
quantifying the asymmetry of quantum states, Nat. Commun. 5, 1
(2014).
[17] F. G. Brandao, M. Horodecki, J. Oppenheim, J. M. Renes, and R.
W. Spekkens, Resource Theory of Quantum States out of Thermal
Equilibrium, Phys. Rev. Lett. 111, 250404 (2013).
[18] V. Veitch, S. H. Mousavian, D. Gottesman, and J. Emerson, The
resource theory of stabilizer quantum computation, New J. Phys. 16,
013009 (2014).
[19] I. Devetak, A. W. Harrow, and A. J. Winter, A resource
framework for quantum shannon theory, IEEE Trans. Inf. Theory 54,
4587 (2008).
[20] B. Eastin and E. Knill, Restrictions on Transversal Encoded
Quantum Gate Sets, Phys. Rev. Lett. 102, 110502 (2009).
[21] Y. Li, M. Han, M. Grassl, and B. Zeng, Invariant perfect
tensors, arXiv preprint arXiv:1612.04504 (2016).
[22] We exclude the case of zero-dimensional Lie groups. Also, G
does not actually need to be a Lie group, but it must have at least
one infinitesimal generator.
[23] M. A. Nielsen and I. Chuang, Quantum computation and quantum
information (2002).
[24] To be precise, each classical gyroscope determines one axis.
In order to send a classical reference frame we need at least two
gyroscopes for the x and y axes. By “gyroscope” we mean a complete
indicator of the reference frame.
[25] Any reader disappointed by the construction’s use of effec-
tively classical gyroscopes should be heartened to know that the
one-into-three encoding described in Appendix A achieves covariant
error correction without them.
[26] Some readers might take issue with calling this an erasure
code, since such codes are usually constructed such that the
Hilbert spaces of each share are the same. However, one can use
infinite-dimensional Hilbert spaces for each share and simply embed
finite dimensional spaces such that the group acts on these
subspaces according to the associated finite-dimensional
representation and trivially on the rest.
[27] In fact, it is not necessary to assume that the basis is
indexed by group elements—they can be indexed by any set on which
the group acts faithfully.
[28] P. W. Shor, in lecture notes, MSRI Workshop on Quantum
Computation (2002), https://www.msri.org/workshops/203/
schedules/1181.
[29] I. Devetak, The private classical capacity and quantum
capacity of a quantum channel, IEEE Trans. Inf. Theory 51, 44
(2005).
[30] S. Lloyd, Capacity of the noisy quantum channel, Phys. Rev. A
55, 1613 (1997).
[31] P. Hayden, M. Horodecki, A. Winter, and J. Yard, A decou-
pling approach to the quantum capacity, Open Syst. Inf. Dyn. 15, 7
(2008).
[32] M. Hamada, Information rates achievable with algebraic codes
on quantum discrete memoryless channels, IEEE Trans. Inf. Theory
51, 4263 (2005).
[33] We thank Beni Yoshida for pointing out the connection to the
Eastin-Knill theorem.
[34] The Eastin-Knill theorem also discusses the possibility of
encoding information in the disconnected components of the Lie
group, a point that is absent in our work. Fur- thermore, the full
Eastin-Knill theorem makes reference to universal gates. In order
to fully reproduce the theorem, we would need additional arguments
concerning continuity of the channel and error detection.
[35] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill,
Holographic quantum error-correcting codes: Toy mod- els for the
bulk/boundary correspondence, arXiv preprint arXiv:1503.06237
(2015).
[36] P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P.
Hayden, and J. Preskill, Continuous symmetries and approximate
quantum error correction, arXiv preprint arXiv:1902.07714
(2019).
[37] M. P. Woods and Á. M. Alhambra, Continuous groups of
transversal gates for quantum error correcting codes from finite
clock reference frames, arXiv preprint arXiv:1902.07725
(2019).
COVARIANT QUANTUM ERROR CORRECTION... PRX QUANTUM 2, 010326
(2021)
[38] S. Zhou, Z.-W. Liu, and L. Jiang, New perspectives on
covariant quantum error correction, arXiv preprint arXiv:2005.11918
(2020).
[39] D.-S. Wang, G. Zhu, C. Okay, and R. Laflamme, Quasi- exact
quantum computation, Phys. Rev. Res. 2, 033116 (2020).
[40] Y. Yang, Y. Mo, J. M. Renes, G. Chiribella, and M. P. Woods,
Covariant quantum error correcting codes via reference frames,
arXiv preprint arXiv:2007.09154 (2020).
[41] P. Hayden, D. W. Leung, and A. Winter, Aspects of generic
entanglement, Commun. Math. Phys. 265, 95 (2006).
[42] A. Harrow, P. Hayden, and D. Leung, Superdense Coding of
Quantum States, Phys. Rev. Lett. 92, 187901 (2004).
[43] P. Hayden, D. Leung, P. W. Shor, and A. Winter, Ran- domizing
quantum states: Constructions and applications, Commun. Math. Phys.
250, 371 (2004).
[44] C. H. Bennett, P. Hayden, D. W. Leung, P. W. Shor, and A.
Winter, Remote preparation of quantum states, IEEE Trans. Inf.
Theory 51, 56 (2005).
IV.. RESULTS
V.. CASE 1: G IS A LIE GROUP AND THE CODE IS FINITE
DIMENSIONAL
VI.. CASE 2: G IS A LIE GROUP AND THE CODE IS INFINITE
DIMENSIONAL
VII.. CASE 3: G IS A FINITE GROUP AND THE CODE IS FINITE
DIMENSIONAL
VIII.. DISCUSSION
. APPENDIX A: G = U(1) AND THE CODE IS CONTINUOUS VARIABLE
. APPENDIX B: G IS A FINITE GROUP AND THE CODE IS A RANDOM
G-COVARIANT ISOMETRY
. APPENDIX C: STEP 1
. APPENDIX D: STEP 2
. APPENDIX E: STEP 3
<< /ASCII85EncodePages false /AllowTransparency false
/AutoPositionEPSFiles true /AutoRotatePages /All /Binding /Left
/CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1)
/CalCMYKProfile () /sRGBProfile (sRGB IEC61966-2.1)
/CannotEmbedFontPolicy /Warning /CompatibilityLevel 1.4
/CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed
true /PassThroughJPEGImages true /CreateJobTicket false
/DefaultRenderingIntent /Default /DetectBlends true /DetectCurves
0.0000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails
false /EmbedAllFonts true /EmbedOpenType false
/ParseICCProfilesInComments true /EmbedJobOptions true
/DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1
/ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 5
/Optimize true /OPM 1 /ParseDSCComments true
/ParseDSCCommentsForDocInfo true /PreserveCopyPage true
/PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness
false /PreserveHalftoneInfo false /PreserveOPIComments false
/PreserveOverprintSettings true /StartPage 1 /SubsetFonts true
/TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue
false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [
true ] /AntiAliasColorImages false /CropColorImages false
/ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK
/DownsampleColorImages true /ColorImageDownsampleType /Average
/ColorImageResolution 300 /ColorImageDepth -1
/ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold
1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode
/AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG
/ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1]
/VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15
/HSamples [1 1 1 1] /VSamples [1 1 1 1] >>
/JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256
/Quality 30 >> /JPEG2000ColorImageDict << /TileWidth
256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false
/CropGrayImages false /GrayImageMinResolution 300
/GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true
/GrayImageDownsampleType /Average /GrayImageResolution 300
/GrayImageDepth -1 /GrayImageMinDownsampleDepth 2
/GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true
/GrayImageFilter /DCTEncode /AutoFilterGrayImages true
/GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict <<
/QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >>
/GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples
[1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth
256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict
<< /TileWidth 256 /TileHeight 256 /Quality 30 >>
/AntiAliasMonoImages false /CropMonoImages false
/MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK
/DownsampleMonoImages true /MonoImageDownsampleType /Average
/MonoImageResolution 1200 /MonoImageDepth -1
/MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true
/MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1
>> /AllowPSXObjects false /CheckCompliance [ /PDFX1a:2003 ]
/PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError false /PDFXTrimBoxToMediaBoxOffset [ 33.84000
33.84000 33.84000 33.84000 ] /PDFXSetBleedBoxToMediaBox false
/PDFXBleedBoxToTrimBoxOffset [ 9.00000 9.00000 9.00000 9.00000 ]
/PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier ()
/PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False
/CreateJDFFile false /Description << /ARA
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
/BGR
<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>
/CHS
<FEFF4f7f75288fd94e9b8bbe5b9a521b5efa7684002000500044004600206587686353ef901a8fc7684c976262535370673a548c002000700072006f006f00660065007200208fdb884c9ad88d2891cf62535370300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c676562535f00521b5efa768400200050004400460020658768633002>
/CHT
<FEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef653ef5728684c9762537088686a5f548c002000700072006f006f00660065007200204e0a73725f979ad854c18cea7684521753706548679c300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002>
/CZE
<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>
/DAN
<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>
/DEU
<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>
/ESP
<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>
/ETI
<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>
/FRA
<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>
/GRE
<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>
/HEB
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
/HRV
<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>
/HUN
<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>
/ITA
<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>
/JPN
<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>
/KOR
<FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020b370c2a4d06cd0d10020d504b9b0d1300020bc0f0020ad50c815ae30c5d0c11c0020ace0d488c9c8b85c0020c778c1c4d560002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/LTH
<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>
/LVI
<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>
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken
voor kwaliteitsafdrukken op desktopprinters en proofers. De
gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe
Reader 5.0 en hoger.) /NOR
<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>
/POL
<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>
/PTB
<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>
/RUM
<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>
/RUS
<FEFF04180441043f043e043b044c04370443043904420435002004340430043d043d044b04350020043d0430044104420440043e0439043a043800200434043b044f00200441043e043704340430043d0438044f00200434043e043a0443043c0435043d0442043e0432002000410064006f006200650020005000440046002c0020043f044004350434043d04300437043d043004470435043d043d044b044500200434043b044f0020043a0430044704350441044204320435043d043d043e04390020043f043504470430044204380020043d04300020043d043004410442043e043b044c043d044b04450020043f04400438043d044204350440043004450020043800200443044104420440043e04390441044204320430044500200434043b044f0020043f043e043b044304470435043d0438044f0020043f0440043e0431043d044b04450020043e0442044204380441043a043e0432002e002000200421043e043704340430043d043d044b04350020005000440046002d0434043e043a0443043c0435043d0442044b0020043c043e0436043d043e00200020043e0442043a0440044b043204300442044c002004410020043f043e043c043e0449044c044e0020004100630072006f00620061007400200438002000410064006f00620065002000520065006100640065007200200035002e00300020043800200431043e043b043504350020043f043e04370434043d043804450020043204350440044104380439002e>
/SKY
<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>
/SLV
<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>
/SUO
<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>
/SVE
<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>
/TUR
<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>
/UKR
<FEFF04120438043a043e0440043804410442043e043204430439044204350020044604560020043f043004400430043c043504420440043800200434043b044f0020044104420432043e04400435043d043d044f00200434043e043a0443043